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The light-harvesting excitonic properties of poly(phenylene ethynylene) (PPE) extended dendrimers (tree-like π-conjugated macromolecules) involve a directional cascade of local excitation energy transfer (EET) processes occurring from the "leaves" (shortest branches) to the "trunk" (longest branch), which can be viewed from a vibronic perspective as a sequence of internal conversions occurring among a connected graph of nonadiabatically coupled locally excited electronic states via conical intersections. The smallest PPE building block that is able to exhibit EET, the asymmetrically meta-substituted PPE oligomer with one acetylenic bond on one side and two parallel ones on the other side (hence, 2-ring and 3-ring para-substituted pseudo-fragments), is a prototype and the focus of the present work. From linear-response time-dependent density functional theory electronic-structure calculations of the molecule as regards its first two nonadiabatically coupled, optically active, singlet excited states, we built a (1 + 2)-state-8-dimensional vibronic-coupling Hamiltonian model for running subsequent multiconfiguration time-dependent Hartree wavepacket relaxations and propagations, yielding both steady-state absorption and emission spectra as well as real-time dynamics. The EET process from the shortest branch to the longest one occurs quite efficiently (about 80% quantum yield) within the first 25 fs after light excitation and is mediated vibrationally through acetylenic and quinoidal bond-stretching modes together with a particular role given to the central-ring anti-quinoidal rock-bending mode. Electronic and vibrational energy relaxations, together with redistributions of quantum populations and coherences, are interpreted herein through the lens of a nonadiabatic perspective, showing some interesting segregation among the foremost photoactive degrees of freedom as regards spectroscopy and reactivity. [ABSTRACT FROM AUTHOR]

We discuss via general arguments and examples when and why the steady nonequilibrium heat capacity vanishes with temperature. The framework is that of Markov jump processes on finite connected graphs where the condition of local detailed balance allows to identify the heat fluxes, and where the discreteness more easily enables sufficient nondegeneracy of the stationary distribution at absolute zero, as under equilibrium. However, for the nonequilibrium extension of the Third Law of Thermodynamics, a dynamic condition is needed as well: the low-temperature dynamical activity and accessibility of the dominant state must remain sufficiently high so that relaxation times do not start to dramatically differ between different initial states. It suffices that the relaxation times do not exceed the dissipation time. [ABSTRACT FROM AUTHOR]

A connected graph G is distance hereditary if every induced path in G is a shortest path. The Interval function I G (u , v) of a connected graph G is defined as the set of all vertices that lie on some shortest u , v -path in G. In this paper, we consider certain types of first-order betweenness axioms framed on an arbitrary function known as transit function and used to characterize the interval function I G of a distance hereditary graph. As a byproduct, we give a new characterization of distance hereditary graphs using these betweenness axioms. [ABSTRACT FROM AUTHOR]

Consider a connected simple graph G. Given a spanning tree T of G , for each edge e in G but not in T , a cycle C e is formed by adding the edge e to the path in T that connects the endpoints of e. The Minimum Spanning Tree Cycle Intersection problem (MSTCI for short) consists in finding a spanning tree for G that minimizes the number of intersections between this type of cycles. This problem was introduced in 2021 and its solution turned out to be difficult for general graphs, without an efficient algorithm to solve it. It was then conjectured that a solution of the problem for a graph that has a universal vertex u is the star centered at u. The conjecture was quickly proven true. In this note, we give a proof of the conjecture that is shorter than the one that has already been published. It is based on a single lemma about domination. We also explore the connections between this lemma and some graph classes, like chordal graphs and dually chordal graphs. [ABSTRACT FROM AUTHOR]

To any simple graph G , the clique graph operator K assigns the graph K (G) , which is the intersection graph of the maximal complete subgraphs of G . The iterated clique graphs are defined by K 0 (G) = G and K n (G) = K (K n - 1 (G)) for n ≥ 1 . We associate topological concepts to graphs by means of the simplicial complex Cl (G) of complete subgraphs of G . Hence, we say that the graphs G 1 and G 2 are homotopic whenever Cl (G 1) and Cl (G 2) are. A graph G such that K n (G) ≃ G for all n ≥ 1 is called K -homotopy permanent. A graph is Helly if the collection of maximal complete subgraphs of G has the Helly property. Let G be a Helly graph. Escalante (1973) proved that K (G) is Helly, and Prisner (1992) proved that G ≃ K (G) , and so Helly graphs are K -homotopy permanent. We conjecture that if a graph G satisfies that K m (G) is Helly for some m ≥ 1 , then G is K -homotopy permanent. If a connected graph has maximum degree at most four and is different from the octahedral graph, we say that it is a low degree graph. It was recently proven that all low-degree graphs G satisfy that K 2 (G) is Helly. In this paper, we show that all low-degree graphs have the homotopy type of a wedge or circumferences, and that they are K -homotopy permanent. [ABSTRACT FROM AUTHOR]

Laplacian matrix and its spectrum are commonly used for giving a measure in networks in order to analyse its topological properties. In this paper, Laplacian matrix of graphs and their spectrum are studied. Laplacian energy of a graph G of order n is defined as LE (G) = ∑ i = 1 n | λ i (L) - d ¯ | , where λ i (L) is the i-th eigenvalue of Laplacian matrix of G, and d ¯ is their average. If LE (G) = LE (K n) for the complete graph K n of order n, then G is known as L-borderenergetic graph. In the first part of this paper, we construct three infinite families of non-complete disconnected L-borderenergetic graphs: Λ 1 = { G b , j , k = [ (((j - 2) k - 2 j + 2) b + 1) K (j - 1) k - (j - 2) ] ∪ b (K j × K k) | b , j , k ∈ Z + } , Λ 2 = { G 2 , b = [ K 6 ∇ b (K 2 × K 3) ] ∪ (4 b - 2) K 9 | b ∈ Z + } , Λ 3 = { G 3 , b = [ b K 8 ∇ b (K 2 × K 4) ] ∪ (14 b - 4) K 8 b + 6 | b ∈ Z + } , where ∇ is join operator and × is direct product operator on graphs. Then, in the second part of this work, we construct new infinite families of non-complete connected L-borderenergetic graphs Ω 1 = { K 2 ∇ a K 2 r ¯ | a ∈ Z + } , Ω 2 = { a K 3 ∪ 2 (K 2 × K 3) ¯ | a ∈ Z + } and Ω 3 = { a K 5 ∪ (K 3 × K 3) ¯ | a ∈ Z + } , where G ¯ is the complement operator on G. [ABSTRACT FROM AUTHOR]

We call a (not necessarily properly) edge-colored graph edge-color-avoiding connected if after the removal of edges of any single color, the graph remains connected. For vertex-colored graphs, similar definitions of color-avoiding connectivity can be given. In this article, we investigate the problem of determining the maximum number of edges that can be removed from either an edge- or a vertex-colored, color-avoiding connected graph so that it remains color-avoiding connected. First, we prove that this problem is NP-hard, and then, we give a polynomial-time approximation algorithm for it. To analyze the approximation factor of this algorithm, we determine the minimum number of edges of color-avoiding connected graphs on a given number of vertices and with a given number of colors. Furthermore, we also consider a generalization of edge-color-avoiding connectivity to matroids. [ABSTRACT FROM AUTHOR]

A connected nontrivial graph G is matching covered , if every edge of G is contained in a perfect matching of G. An edge e of a matching covered graph G is removable if G − e is still matching covered. Lovász and Plummer introduced removable edges in connection with ear decompositions of matching covered graphs. By characterizing the structure of removable edges, we obtain that the number of removable edges of Halin graphs G with even number of vertices, other than K 4 , C 6 ¯ and R 8 , is at least 1 4 (| V (G) | + 2) , and the lower bound is attainable. [ABSTRACT FROM AUTHOR]

In order to meet ever-increasing demands for reliable parallel and distributed systems, it is crucial to evaluate the reliability and fault tolerance of their underlying interconnection networks. Such an interconnection network is usually modeled as a connected graph G, where the vertex set and edge set represent the processors and links between processors in the network, respectively. In this paper, we combine Fàbrega's idea about h-extra edge-connectivity and Sampathkumar's concept about r-component edge-connectivity to introduce a more refined parameter for characterizing fault tolerance of interconnection networks, named as h-extra r-component edge-connectivity. Given a connected graph G, for two integers h ≥ 1 and r ≥ 2 , the h-extra r-component edge-connectivity of G, denoted as c λ r h (G) , is the minimum cardinality among all edge subsets F ⊂ E (G) , if any, such that G - F has at least r components, and each component has at least h vertices. As an enhancement on hypercube, the n-dimensional augmented cube AQ n , introduced by Choudum and Sunitha in 2002, reserves several excellent topological properties. As | V (AQ n) | = 2 n , the h-extra three-component edge-connectivity of AQ n is well-defined for each integer h with 1 ≤ h ≤ ⌊ 2 n / 3 ⌋ . In this paper, a generalization of Xu et al.'s conclusion is obtained that finds an upper bound for the exact value of general h-extra three-component edge-connectivity of AQ n and shows that it is sharp for 1 ≤ h ≤ 2 n 2 - 1 and h = 2 c where 1 ≤ c ≤ n - 2 . Let h = ∑ i = 0 s 2 t i be a positive integer with t 0 > t 1 > ⋯ > t s ≥ 0 . Let δ = 0 if h is even and δ = 1 if h is odd. Specifically, c λ 3 h (AQ n) = (4 n - 4) h - 2 ∑ i = 0 s (2 t i - 1) 2 t i - 2 ∑ i = 0 s 4 i · 2 t i - δ for n ≥ 4 , h ≤ 2 n 2 - 1 , and c λ 3 2 c (AQ n) = (2 n - 2 c - 1) 2 c + 1 for n ≥ 4 and 1 ≤ c ≤ n - 2 . [ABSTRACT FROM AUTHOR]

We study in this paper single-peakedness on arbitrary graphs. Given a collection of preferences (rankings of alternatives), we aim at determining a connected graph G on which the preferences are single-peaked, in the sense that all the preferences are traversals of G. Note that a collection of preferences is always single-peaked on the complete graph. We propose an Integer Linear Programming formulation (ILP) of the problem of minimizing the number of edges in G or the maximum degree of a vertex in G. We prove that both problems are NP-hard in the general case. However, we show that if the optimal number of edges is m − 1 (where m is the number of candidates) then any optimal extreme point solution of the continuous relaxation of the ILP is integer and thus the integrality constraints can be relaxed. This provides an alternative proof of the polynomial time complexity of recognizing single-peaked preferences on a tree. We prove the same result for the case of a path (an axis), providing here also an alternative proof of polynomiality of the recognition problem. Furthermore, we provide a polynomial time procedure to recognize single-peaked preferences on a pseudotree (a connected graph that contains at most one cycle). We also give some experimental results, both on real and synthetic datasets. [ABSTRACT FROM AUTHOR]

A cactus is a connected graph in which each edge is contained in at most one cycle. We generalize the concept of cactus graphs, i.e., a k -cactus is a connected graph in which each edge is contained in at most k cycles where k ≥ 1. It is well known that every cactus with n vertices has at most ⌊ 3 2 (n − 1) ⌋ edges. Inspired by it, we attempt to establish analogous upper bounds for general k -cactus graphs. In this paper, we first characterize k -cactus graphs for 2 ≤ k ≤ 4 based on the block decompositions. Subsequently, we give tight upper bounds on their sizes. Moreover, the corresponding extremal graphs are also characterized. However, for larger k , this extremal problem remains unsolved. Besides, we prove that every 2-connected k -cactus (k ≥ 1) with n vertices has at most n + k − 1 edges, and the bound is tight if n ≥ k + 2. [ABSTRACT FROM AUTHOR]

A profile π = (x 1 , x 2 , ... , x k) of length k in a connected graph G is a sequence of vertices in G with possible repetitions of vertices. A median x of a profile π in G is a vertex that minimizes the remoteness value, that is, the sum of the distances from x to the elements x i in π is minimized. The median function output the set of medians (denoted as M e d (π)) of π , for every profile π in G. It is one of the basic models for the location of a desirable facility in a network. The median function on graphs is well studied. For instance, it has been characterized axiomatically by simple axioms on trees, hypercubes, median graphs, cocktail-party graphs, and complete graphs minus a matching. In this paper, we study the median sets and median function of block graphs, an immediate generalization of trees. We determine the median sets of all types of profiles and we obtain two sets of axiomatic characterizations of the median function on block graphs. [ABSTRACT FROM AUTHOR]

A graph G is said to be strongly Menger edge-connected (SM- λ for short) if for every two of its vertices u and v , there are min { d G (u) , d G (v) }) edge-disjoint paths between them, where d G (u) is the degree of u in G. The maximum conditional edge-fault-tolerant number of order r with respect to the SM- λ property of G , denoted by s m λ r (G) , is the maximum integer m such that G − F remains SM- λ for every edge set F with | F | ≤ m and δ (G − F) ≥ r. So far, most of the exact values of s m λ r (G) are for the case r ≤ 2 for special class of graphs. In this paper, we give a sufficient condition to give a bound of s m λ r (G) for general r on regular graphs. As an application, we determine the exact value of s m λ 2 r − 1 (A Q n) for 1 ≤ r ≤ n − 2 where A Q n is the n -dimensional augmented cube. [ABSTRACT FROM AUTHOR]

Alishahi and Hajiabolhassan found that for some classes of graphs, the wonderful equality χ G , r K 2 = | E (G) | − ex G , r K 2 holds as an amazing relationship between chromatic number and generalized Turán number. This powerful equality enabled them to determine chromatic numbers of some interesting and important classes of graphs. They conjectured that the aforementioned equality holds for all connected graphs G. Iradmusa, by a nice elegant use of a class of cubic graphs, called snarks, made counterexamples to this conjecture for which χ G , r K 2 = 1 and | E (G) | − ex G , r K 2 = 3. In this paper, for any arbitrary positive integer Θ , we explicitly construct a sequence of trees T r r = 1 ∞ for which lim r → ∞ (| E T r | − ex T r , r K 2 ) = + ∞ while χ T r , r K 2 = Θ for all r ≥ 3. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph. The eccentric distance sum (EDS) of G is defined as ξ d (G) = ∑ v ∈ V (G) ɛ G (v) D G (v) , where ɛ G (v) is the eccentricity of the vertex v and D G (v) = ∑ u ∈ V (G) d G (u , v) is the sum of all distances from the vertex v. We denote the set of trees with order n and maximum degree Δ by T n , Δ. In 2015, the tree having the maximal EDS among all trees in T n , Δ was determined (Miao, 2015). In this paper, the tree having the second maximal EDS among all trees in T n , Δ is characterized. [ABSTRACT FROM AUTHOR]

Given a connected graph G = (V , E) , with n ≔ | V | vertices and m ≔ | E | edges, a cut can be represented as a bipartition { S , S ¯ } of the vertices or as the set of those edges in E having one endpoint in S and the other in S ¯ , denoted by δ G (S , S ¯). A cut is minimal, or also called bond , if and only if the two induced subgraphs obtained by removing the edges in the cut are both connected. When the bond separates two given vertices s and t , we talk about s , t - bond. In this work, we consider the problems of listing all the bonds and listing all the s , t -bonds in a graph. These fundamental problems find application in many research areas, such as, beyond graph theory, network reliability, bioinformatics, and chemistry. The state-of-the-art algorithm exploits binary partition to output each s , t -bond in O (m) per bond, being thus classified as an O (m) -delay algorithm. Here we present two new algorithms to address these problems. The first one implements a slightly different branching strategy than the state of the art, though achieving the same complexity. Anyway, we can improve it by relying on dynamic data structures and amortized analysis, obtaining an algorithm that outputs a new bond in O ˜ (n). By assuming only the two bond-shores are output for every bond, this is the first output-linear algorithm listing bonds. In case we commit to explicitly providing the entire edge-set of every bond, the delay becomes O ˜ (n) + | δ G (S , S ¯) |. • We study the problem of listing all minimal cuts (or bonds) in a graph. • We exploit a different branching strategy than Tsukiyama et al. (1980). • We provide a first algorithm having the same complexity as Tsukiyama et al. (1980). • We rely on dynamic data structures and amortized analysis to improve our approach. • We propose the first O ˜ (n) -delay algorithm to list all bonds in a graph. [ABSTRACT FROM AUTHOR]

Prescribed performance control allows preselected transient and steady‐state bounds of formation control performance. However, owing to the predetermined decaying property of performance functions, it is unavailable for the formation tracking problem with dynamic obstacle avoidance. Thus, it is essential to design a flexible performance function to accommodate the increasing formation error during dynamic obstacle avoidance. This article pays attention to the development of an adaptive flexible performance (AFP) function usable for the avoidance of dynamic obstacles with unknown velocities in the distributed formation framework. We develop an AFP‐based distributed formation tracker design for range‐constrained switched multi‐input multi‐output nonlinear multiagent systems with asynchronous switching. The AFP functions for ensuring graph connectivity and dynamic obstacle avoidance are derived by designing time‐varying variables adjusted by an adaptive relaxation signal. The neural‐network‐based adaptive distributed formation tracker using the derived performance functions is constructed with the adaptive relaxation signal to ensure the stability of the closed‐loop formation system, regardless of asynchronous switching and dynamic obstacles. The effectiveness of the proposed method is shown by simulation results. [ABSTRACT FROM AUTHOR]

This paper proposes a novel free-weighting matrix (FWM) approach for the event-triggered (ET) consensus and formation control of multi-agent systems over a directed graph. A new FWM equality for the consensus error is proposed. This FWM is applied to attain coherent behavior of systems via a fully distributed ET approach. The proposed approach is also extended for ET formation control to attain a specific pattern of agents. Furthermore, an FWM-based consensus controller design with practical ET cooperative control is considered to address consensus and formation control problems by using the in-degree information of a node rather than the algebraic connectivity of a graph or adaptation of parameters. Additionally, the controller gain is formulated using a more general transformation compared to that used in classical approaches. Moreover, the proposed ET scheme for the designed controller is applied on the transmission side to preserve computational resources by eliminating the Zeno behavior. In contrast to the existing methods, the proposed approach (i) provides a fully distributed protocol without using central information of network or adaptive gain; (ii) provides a practical ET mechanism to assure effective bandwidth and fewer energy consumption; and (iii) can be applied to the ET formation problem for mobile agents. Simulations involving six aerial vehicles are performed to demonstrate the efficacy of the proposed cooperative control strategy. [ABSTRACT FROM AUTHOR]

Let s and t be positive integers, and let G be a simple and connected graph with vertex set V. For x , y ∈ V , let d (x , y) denote the length of an x − y geodesic in G and let d s (x , y) = min { d (x , y) , s + 1 }. Let W ⊆ V. A set W is a distance- s locating set of G if, for any distinct x , y ∈ V , there exists a vertex z ∈ W such that d s (x , z) ≠ d s (y , z) , and the distance- s location number, dim s (G) , of G is the minimum cardinality among all distance- s locating sets of G. A set W is a distance- t dominating set of G if, for each vertex u ∈ V , there exists a vertex v ∈ W such that d (u , v) ≤ t , and the distance- t domination number, γ t (G) , of G is the minimum cardinality among all distance- t dominating sets of G. The (s , t) -location-domination number of G , denoted by γ L (s , t) (G) , is the minimum cardinality among all sets W ⊆ V such that W is both a distance- s locating set and a distance- t dominating set of G. For any connected graph G of order at least two, we show that 1 ≤ max { dim s (G) , γ t (G) } ≤ γ L (s , t) (G) ≤ min { dim α (G) + 1 , | V | − 1 } , where α = min { s , t }. We characterize connected graphs G satisfying γ L (s , t) (G) equals 1 and | V | − 1 , respectively. We examine the relationship among dim s (G) , γ t (G) and γ L (s , t) (G) ; along the way, we show that γ L (s , t) (G) = dim s (G) if s < t. We also show that there exist graphs H and G with H ⊂ G such that γ L (s , t) (H) γ L (s , t) (G) can be arbitrarily large. Moreover, we examine some graph classes. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a connected graph where V is the set of vertices of G and E is the set of edges of G. The distance from the vertex w ∈ V to the edge  e = u v ∈ E is given by d (e , w) = min { d (u , w) , d (v , w) }. A subset S e ⊂ V is called an edge metric generator for G if for every two distinct edges e 1 , e 2 ∈ E , there exists a vertex w ∈ S such that d (e 1 , w) ≠ d (e 2 , w). The edge metric generator with the minimum number of vertices is called an edge metric basis for G and the cardinality of the edge metric basis is called the edge metric dimension denoted by dim e (G). A subset S m ⊂ V is called a mixed metric generator for G if for every two distinct elements x , y ∈ V ∪ E , there exists a vertex w ∈ S such that d (x , w) ≠ d (y , w). A mixed metric generator containing a minimum number of vertices is called a mixed metric basis for G and the cardinality of a mixed metric basis is called the mixed metric dimension denoted by dim m (G). In this paper, we study the edge metric dimension and the mixed metric dimension of a plane graph  T n . [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph. A configuration of pebbles is a function from V (G) to a set of integers. Consider a configuration of pebbles on G. A pebbling move consists of removing two pebbles off a vertex and placing one on an adjacent vertex. The monophonic pebbling number of G is the smallest integer μ (G) from which we can put one pebble to a target using monophonic path through pebbling moves. The monophonic t -pebbling number of G is the smallest positive integer μ t (G) such that from any configuration of μ t (G) pebbles we can put t pebbles on a target using monophonic path. Here we discuss these concepts for families of cycles. [ABSTRACT FROM AUTHOR]

The fault tolerability of the network is usually measured by the classical or generalized connectivity of the graph. For any subset $ S \subseteq V\left (G \right) $ S ⊆ V (G) with $ \left | S \right | \ge 2 $ | S | ≥ 2 , a tree T is called an S-tree if $ S \subseteq V\left (T \right) $ S ⊆ V (T) . Furthermore, any two S-tree $ T_1 $ T 1 and $ T_2 $ T 2 are internally disjoint if $ E\left ({{T_1}} \right) \cap E\left ({{T_2}} \right) = \emptyset $ E ( T 1 ) ∩ E ( T 2 ) = ∅ and $ V\left ({{T_1}} \right) \cap V\left ({{T_2}} \right) = S $ V ( T 1 ) ∩ V ( T 2 ) = S. We denote by $ {\kappa _G}\left (S \right) $ κ G (S) the maximum number of pairwise internally disjoint S-trees in G. For an integer $ k \ge 2 $ k ≥ 2 , the generalized k-connectivity of a graph G is defined as $ {\kappa _k}\left (G \right) = \min \left \{ {{\kappa _G}\left (S \right) | S \subseteq V\left (G \right)} \right. $ κ k (G) = min { κ G (S) | S ⊆ V (G) and $ \left. {\left | S \right | = k} \right \} $ | S | = k } . In this paper, we establish the generalized 4-connectivity of the Cayley graph $ CT_n $ C T n generated by complete graphs. [ABSTRACT FROM AUTHOR]

A total dominating set in a graph G $G$ is a set of vertices of G $G$ such that every vertex is adjacent to a vertex of the set. The total domination number γt(G) ${\gamma }_{t}(G)$ is the minimum cardinality of a total dominating set in G $G$. In this paper, we study the following open problem posed by Yeo. For each Δ≥3 ${\rm{\Delta }}\ge 3$, find the smallest value, rΔ ${r}_{{\rm{\Delta }}}$, such that every connected graph G $G$ of order at least 3, of order n $n$, size m $m$, total domination number γt ${\gamma }_{t}$, and bounded maximum degree Δ ${\rm{\Delta }}$, satisfies m≤12(Δ+rΔ)(n−γt) $m\le \frac{1}{2}({\rm{\Delta }}+{r}_{{\rm{\Delta }}})(n-{\gamma }_{t})$. Henning showed that rΔ≤Δ ${r}_{{\rm{\Delta }}}\le {\rm{\Delta }}$ for all Δ≥3 ${\rm{\Delta }}\ge 3$. Yeo significantly improved this result and showed that 0.1ln(Δ)

In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with n vertices and of valence d , d ≤ 4 , is at most c d n where c 3 = 1 and c 4 = 9. Whether such a constant c d exists for valencies larger than 4 remains an unanswered question. Further, we prove that every automorphism g of a finite connected 3-valent vertex-transitive graph Γ, Γ ≇ K 3 , 3 , has a regular orbit, that is, an orbit of 〈 g 〉 of length equal to the order of g. Moreover, we prove that in this case either Γ belongs to a well understood family of exceptional graphs or at least 5/12 of the vertices of Γ belong to a regular orbit of g. Finally, we give an upper bound on the number of orbits of a cyclic group of automorphisms C of a connected 3-valent vertex-transitive graph Γ in terms of the number of vertices of Γ and the length of a longest orbit of C. [ABSTRACT FROM AUTHOR]

A simple graph G with maximum degree Δ is overfull if | E (G) | > Δ ⌊ | V (G) | / 2 ⌋. The core of G , denoted G Δ , is the subgraph of G induced by its vertices of degree Δ. Clearly, the chromatic index of G equals Δ + 1 if G is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if G is a simple connected graph with Δ ≥ 3 and Δ (G Δ) ≤ 2 , then χ ′ (G) = Δ + 1 implies that G is overfull or G = P ⁎ , where P ⁎ is obtained from the Petersen graph by deleting a vertex. Cariolaro and Cariolaro settled the base case Δ = 3 in 2003, and Cranston and Rabern proved the next case, Δ = 4 , in 2019. In this paper, we give a proof of this conjecture for all Δ ≥ 4. [ABSTRACT FROM AUTHOR]

We consider two types of matroids defined on the edge set of a graph G : count matroids M k , ℓ (G) , in which independence is defined by a sparsity count involving the parameters k and ℓ , and the C 2 1 -cofactor matroid C (G) , in which independence is defined by linear independence in the cofactor matrix of G. We show, for each pair (k , ℓ) , that if G is sufficiently highly connected, then G − e has maximum rank for all e ∈ E (G) , and the matroid M k , ℓ (G) is connected. These results unify and extend several previous results, including theorems of Nash-Williams and Tutte (k = ℓ = 1), and Lovász and Yemini (k = 2 , ℓ = 3). We also prove that if G is highly connected, then the vertical connectivity of C (G) is also high. We use these results to generalize Whitney's celebrated result on the graphic matroid of G (which corresponds to M 1 , 1 (G)) to all count matroids and to the C 2 1 -cofactor matroid: if G is highly connected, depending on k and ℓ , then the count matroid M k , ℓ (G) uniquely determines G ; and similarly, if G is 14-connected, then its C 2 1 -cofactor matroid C (G) uniquely determines G. We also derive similar results for the t -fold union of the C 2 1 -cofactor matroid, and use them to prove that every 24-connected graph has a spanning tree T for which G − E (T) is 3-connected, which verifies a case of a conjecture of Kriesell. [ABSTRACT FROM AUTHOR]

Attention deficit hyperactivity disorder (ADHD) is a common neurodevelopmental disorder that is characterized by inattention, hyperactivity, and impulsivity. The neural mechanisms underlying ADHD remain inadequately understood, and current approaches do not well link neural networks and attention networks within brain networks. Our objective is to investigate the neural mechanisms related to attention and explore neuroimaging biological tags that can be generalized within the attention networks. In this paper, we utilized resting-state functional magnetic resonance imaging data to examine the differential functional connectivity network between ADHD and typically developing individuals. We employed a graph convolutional neural network model to identify individuals with ADHD. After classification, we visualized brain regions with significant contributions to the classification results. Our results suggest that the frontal, temporal, parietal, and cerebellar regions are likely the primary areas of dysfunction in individuals with ADHD. We also explored the relationship between regions of interest and attention networks, as well as the connection between crucial nodes and the distribution of positively and negatively correlated connections. This analysis allowed us to pinpoint the most discriminative brain regions, including the right orbitofrontal gyrus, the left rectus gyrus and bilateral insula, the right inferior temporal gyrus and bilateral transverse temporal gyrus in the temporal region, and the lingual gyrus of the occipital lobe, multiple regions of the basal ganglia and the upper cerebellum. These regions are primarily involved in the attention executive control network and the attention orientation network. Dysfunction in the functional connectivity of these regions may contribute to the underlying causes of ADHD. [ABSTRACT FROM AUTHOR]

This article proposes a new distributed finite‐time optimization algorithm for agents under directed graphs. By employing the nonsmooth technique and graph theory, a distributed discontinuous algorithm for continuous‐time agents subject to strongly convex local cost functions is first designed with a finite‐time distributed estimator, where the gradients of the local cost functions are estimated in finite time. It is shown that for a strongly connected graph and arbitrary initial conditions, the proposed algorithms can achieve consensus, and the systems can converge to the optimal point in finite time. Then, a two‐step approach is proposed to achieve finite‐time optimization of high‐order agents with disturbances under directed graphs. Finally, the validity of the proposed finite‐time optimization algorithm is verified by two numerical examples. [ABSTRACT FROM AUTHOR]

Summary: The connectivity of communication graph is indispensable for consensus of multi‐agent systems (MASs), which in many applications (e.g., wireless sensor) depends on the relative distances between agents. But, in the adaptive setting particularly with system nonlinearities and event‐triggered communication, it is rather difficult to enforce the relative distances within the limited range for the connectivity. This paper focuses on developing an adaptive event‐triggered control strategy with connectivity preservation in the context of nonidentical unknown control coefficients and heterogeneous nonlinearities coupling with parameter uncertainties. First, a group of potential functions are introduced acting as control barrier functions to constrain the relative distances between agents within the limited range for all time. Also, two dynamic gains are specialized for each agent to handle the system uncertainties, system nonlinearities and negative effect of the execution error. Then, an adaptive event‐triggered protocol is designed for each agent such that the connectivity‐preserving consensus of MASs is achieved and Zeno behavior is excluded. Moreover, an extended study is conducted on a leader‐following scenario. Two simulation examples illustrate the effectiveness of the proposed event‐triggered control strategy. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with the adjacency matrix A (G) , and let τ (G) denote the smallest positive eigenvalue of A (G). Let G n be the class of all connected bipartite graphs on n = 2 k vertices with a unique perfect matching. In this article, we characterize the graphs G in G n such that τ (G) does not exceed 1 2. Using the above characterization, we obtain the unique graphs in G n with the maximum and the second maximum τ , respectively. Further, we prove that the largest and the second largest limit points of the smallest positive eigenvalues of bipartite graphs with a unique perfect matching are 1 2 and the reciprocal of α 3 1 2 + α 3 − 1 2 , respectively, where α 3 is the largest root of x 3 − x − 1. [ABSTRACT FROM AUTHOR]

In the Maker–Breaker vertex colouring game, first publicised by Gardner in 1981, Maker and Breaker alternately colour vertices of a graph using a fixed palette, maintaining a proper colouring at all times. Maker aims to colour the whole graph, and Breaker aims to make some vertex impossible to colour. We are interested in the following question, first asked by Zhu in 1999: if Maker wins with k colours available, must they also win with k + 1 ? This question has remained open, attracting significant attention and being reposed for many similar games. While we cannot resolve this problem for the vertex colouring game, we can answer it in the affirmative for the game of arboricity, resolving a question of Bartnicki, Grytczuk, and Kierstead from 2008. We then consider how one might approach the question of monotonicity for the vertex colouring game, and work with a related game in which the vertices must be coloured in a prescribed order. We demonstrate that this 'ordered vertex colouring game' does not have the above monotonicity property, and discuss the implications of this fact to the unordered game. Finally, we provide counterexamples to two open problems concerning a connected version of the graph colouring game. [ABSTRACT FROM AUTHOR]

A new impartial game on a connected graph was introduced, called a feedback game , which is a variant of generalized geography. In this paper, we study the feedback game on 3-chromatic Eulerian triangulations of surfaces. We prove that the winner of the game on every 3-chromatic Eulerian triangulation of a surface all of whose vertices have degree 0 modulo 4 is always fixed. Moreover, we also study the case of 3-chromatic Eulerian triangulations of surfaces which have at least two vertices whose degrees are 2 modulo 4. In addition, as a concrete class of such graphs, we consider the octahedral path, which is obtained from an octahedron by adding octahedra in the same face, and completely determine the winner of the game on those graphs. [ABSTRACT FROM AUTHOR]

For a connected graph G , we denote by m G (μ) , c (G) and p (G) the eigenvalue multiplicity of μ in G , the cyclomatic number and the number of pendant vertices in G , respectively. In 2020, Wang et al. (2020) proved that m G (μ) ≤ 2 c (G) + p (G) for any μ ∈ R , the equality holds if and only if G is a cycle and μ is a multiple eigenvalue of the cycle. We find that when ∥ μ ∥ ≥ 2 , the bound 2 c (G) + p (G) is much rough. Thus we are motivated to improve the bound of m G (μ) from 2 c (G) + p (G) to c (G) + q (G) for the case when ∥ μ ∥ ≥ 2 , where q (G) is the number of quasi-pendant vertices of G. [ABSTRACT FROM AUTHOR]

Graph pebbling is a combinatorial game played on an undirected graph with an initial configuration of pebbles. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The pebbling number of a graph is the smallest number of pebbles necessary such that, given any initial configuration of pebbles, at least one pebble can be moved to a specified root vertex. Recent lines of inquiry apply computational techniques to pebbling bound generation and improvement. Along these lines, we present a computational framework that produces a set of tree strategy weight functions that are capable of proving pebbling number upper bounds on a connected graph. Our mixed-integer linear programming approach automates the generation of large sets of such functions and provides verifiable certificates of pebbling number upper bounds. The framework is capable of producing verifiable pebbling bounds on any connected graph, regardless of its structure or pebbling properties. We apply the model to the 4th weak Bruhat to prove π (B 4) ≤ 66 and to the Lemke square graph to produce a set of certificates that verify π (L □ L) ≤ 96. [ABSTRACT FROM AUTHOR]

The restricted edge-connectivity of a connected graph G, denoted by λ ′ (G) , if it exists, is the minimum cardinality of a set of edges whose deletion makes G disconnected, and each component has at least two vertices. It was proved that λ ′ (G) exists if and only if G has at least four vertices and G is not a star. In this case, a graph G is called maximally restricted edge-connected if λ ′ (G) = ξ (G) , and a graph G is called super restricted edge-connected if each minimum restricted edge-cut isolates an edge of G. The strong product of graphs G and H, denoted by G ⊠ H , is the graph with the vertex set V (G) × V (H) and the edge set { (x 1 , y 1) (x 2 , y 2) | x 1 = x 2 and y 1 y 2 ∈ E (H) ; or y 1 = y 2 and x 1 x 2 ∈ E (G) ; or x 1 x 2 ∈ E (G) and y 1 y 2 ∈ E (H) }. In this paper, we determine, for any nontrivial connected graph G, the restricted edge-connectivity of G ⊠ P n , G ⊠ C n and G ⊠ K n , where P n , C n and K n are the path, cycle and complete graph of order n, respectively. As corollaries, we give sufficient conditions for these strong product graphs G ⊠ P n , G ⊠ C n and G ⊠ K n to be maximally restricted edge-connected and super restricted edge-connected. [ABSTRACT FROM AUTHOR]

Migration flocks have different forms, including single individuals, formations, and irregular clusters. The shape of a flock can change swiftly over time. The real-time clustering of multiple groups with different characteristics is crucial for the monitoring of dynamically changing migratory flocks. Traditional clustering algorithms need to set various prior parameters, including the number of groups, the number of nearest neighbors, or the minimum number of individuals. However, flocks may display complex group behaviors (splitting, combination, etc.), which complicate the choice and adjustment of the parameters. This paper uses a real-time clustering-based method that utilizes concepts from the algebraic graph theory. The connected graph is used to describe the spatial relationship between the targets. The similarity matrix is calculated, and the problem of group clustering is equivalent to the extraction of the partitioned matrices within. This method needs only one prior parameter (the similarity distance) and is adaptive to the group's splitting and combination. Two modifications are proposed to reduce the computation burden. First, the similarity distance can be broadened to reduce the exponent of the similarity matrix. Second, the omni-directional measurements are divided into multiple sectors to reduce the dimension of the similarity matrix. Finally, the effectiveness of the proposed method is verified using the experimental results using real radar data. [ABSTRACT FROM AUTHOR]

Generally, performing multiple scans is necessary to cover entire scanning area, and multiple point clouds are thus obtained. These point clouds need to be registered together, which is the focus of our work. However, influenced by the noise, point density variation, partial overlap and so on, the success rate of registration is still low. For this reason, a pairwise registration method is first proposed. It includes two main components: the triple local coordinate image (TriLCI) descriptor and a new 3D transformation estimation technique. The descriptor has high descriptiveness and strong robustness, by which more right correspondences can be achieved. The 3D transformation estimation technique is employed to calculate the rotation matrix and translation vector under the condition that most of correspondences are false. In this technique, a new geometric distance constraint is developed, and is then applied to eliminate incorrect correspondences. The robust estimation based on the median is introduced to calculate the rotation matrix and translation vector. For multi-view registration, the connected graph algorithm and shape growing based method are applied. A comparative study is presented. The experimental results well demonstrate that the proposed pairwise registration method has high success rate. The comparative study gives the merits and demerits of the two multi-view registration methods. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the construction of symmetric matrix whose corresponding graph is connected and unicyclic using some pre-assigned spectral data. Spectral data for the problem consist of the smallest and the largest eigenvalues of each leading principal submatrices. Inverse eigenvalue problem (IEP) with this set of spectral data is generally known as the extremal IEP. We use a standard scheme of labeling the vertices of the graph, which helps in getting a simple relation between the characteristic polynomials of each leading principal submatrix. Sufficient condition for the existence of the solution is obtained. The proof is constructive, hence provides an algorithmic procedure for finding the required matrix. Furthermore, we provide the condition under which the same problem is solvable when two particular entries of the required matrix satisfy a linear relation. [ABSTRACT FROM AUTHOR]

Fault diameter is an important parameter to measure the reliability and effectiveness of interconnection networks. Strong product is an efficient method to construct large graphs from small graphs. In this paper, we study the fault diameter of strong product graph of an arbitrary connected graph and a complete graph. According to the classification of an arbitrary connected graph, we first determine the fault diameter of strong product graph of two complete graphs. Then we give the fault diameter of strong product graph of an incompletely connected graph and a complete graph, which can be denoted by the fault diameter of its incompletely factor graph. In addition, we also give a more general result about fault diameter. [ABSTRACT FROM AUTHOR]

Consider the graph with the residue classes modulo n as vertices, and the following edges: "additive" edges from a to a + 1; "multiplicative" edges from a to ab for some fixed b. This graph illustrates a criterion for divisibility by n for numbers written in base b. By varying n and b, we see a great variety of structures: this topic connects arithmetic to graph theory and has the beauty of string art. For divisibility graphs we investigate the existence of triangles and other cycles, the girth, the minimum/maximum of the vertex degree, the chromatic number, and the planarity. [ABSTRACT FROM AUTHOR]

Let G be a graph with no isolated vertex. A semitotal forcing set of G is a (zero) forcing set S such that every vertex in S is within distance 2 of another vertex of S. The semitotal forcing number $F_{t2}(G)$ is the minimum cardinality of a semitotal forcing set in G. In this paper, we prove that it is NP-complete to determine the semitotal forcing number of a graph. We also prove that if $G\neq K_n$ is a connected graph of order $n\geq 4$ with maximum degree $\Delta \geq 2$ , then $F_{t2}(G)\leq (\Delta-1)n/\Delta$ , with equality if and only if either $G=C_{4}$ or $G=P_{4}$ or $G=K_{\Delta ,\Delta }$. [ABSTRACT FROM AUTHOR]

In this article, we introduce a novel concept called Generalized Transmission Neighbor Indices, building upon established transmission indices. The primary focus is on two variants of these indices, denoted as T N (a , b) 1 (G) and T N (a , b) 2 (G) , which offer distinct insights into graph connectivity. The first index, T N (a , b) 1 (G) , quantifies the sum of powered vertex neighbor transmissions for connected vertices, while the second, T N (a , b) 2 (G) , calculates the product of powered vertex neighbor transmissions among connected vertices. Our investigation delves into the diverse values of parameters a and b, shedding light on the relationships between these indices and established transmission neighbor-based metrics. Bounds have been computed, and we have also explored the chemical relevance (Quantitative Structure–Property Relationship) in the context of linear monocarboxylic acids. [ABSTRACT FROM AUTHOR]

The distance energy of a connected graph G is the sum of absolute values of the eigenvalues of the distance matrix of G. In this paper, we study how the distance energy of the complete split graph G S (m , n) = K m + K ¯ n changes when an edge is deleted from it. The complete split graph G S (m , n) consists of a clique on m vertices and an independent set on n vertices in which each vertex of the clique is adjacent to each vertex of the independent set. We prove that the distance energy of the complete split graph G S (m , n) always increases when an edge is deleted from it. [ABSTRACT FROM AUTHOR]

Let be a simple connected graph consisting of n vertices and m edges. In a proper l -coloring, an r -dynamic coloring of a graph is one in which each vertex's neighbors are provided at least min { r , d (k) } different colors. The r -dynamic chromatic number of graph , given as χ r () , is the minimal l such that the graph has r -dynamic l coloring of . In this study, we combine a few common graphs to provide the r -dynamic coloring of the subdivision-edge neighborhood corona of path graph P s and star graph K 1 , s . These graphs include path graph P q , cycle graph C q , complete graph K q , star graph K 1 , q , and double star graph K 1 , q , q . [ABSTRACT FROM AUTHOR]

Let G be a finite undirected simple connected graph with vertex set V (G) and edge set E (G). A vertex u ∈ V (G) resolves two elements (vertices or edges) v , w ∈ V (G) ∪ E (G) if d (u , v) ≠ d (u , w). A subset S m of vertices in G is called a mixed metric generator for G if every two distinct elements (vertices and edges) of G are resolved by some vertices of S m . The minimum cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by dim m (G). In this paper, we study the mixed metric dimension for the plane graph of web graph n and convex polytope n . [ABSTRACT FROM AUTHOR]

Let G be a connected graph on n vertices with adjacency matrix A G. Associated to G is a polynomial d G (x 1 , ... , x n) of degree n in n variables, obtained as the determinant of the matrix M G (x 1 , ... , x n) , where M G = Diag (x 1 , ... , x n) − A G. We investigate in this article the set V d G (r) of non-negative values taken by this polynomial when x 1 , ... , x n ≥ r ≥ 1. We show that V d G (1) = Z ≥ 0. We show that for a large class of graphs one also has V d G (2) = Z ≥ 0. When V d G (2) ≠ Z ≥ 0 , we show that for many graphs V d G (2) is dense in Z ≥ 0. We give numerical evidence that in many cases, the complement of V d G (2) in Z ≥ 0 might in fact be finite. As a byproduct of our results, we show that every graph can be endowed with an arithmetical structure whose associated group is trivial. [ABSTRACT FROM AUTHOR]