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Nikiforov defined the A α -matrix of a graph G as A α (G) = α D (G) + (1 − α) A (G) , where α ∈ [ 0 , 1 ] , D (G) and A (G) are the diagonal matrix of degrees and the adjacency matrix respectively. The largest eigenvalue of A α (G) is called the A α -spectral radius of G , denoted by ρ α (G). In this paper, we first give an upper bound on ρ α (G) of a connected graph G with fixed size m ≥ 3 k and maximum degree Δ ≤ m − k , where k is a positive integer. For two connected graphs G 1 and G 2 with size m ≥ 4 , employing this upper bound, we prove that ρ α (G 1) > ρ α (G 2) if Δ (G 1) > Δ (G 2) and Δ (G 1) ≥ 2 m 3 + 1. As an application, we determine the graph with the maximal A α -spectral radius among all graphs with fixed size and girth. Our theorems generalize the recent results for the signless Laplacian spectral radius of a graph. [ABSTRACT FROM AUTHOR]

Identifying the graph with maximum or minimum spectral radius among a given class of graphs is a central problem in extremal spectral graph theory, known as the Brualdi–Solheid problem. For a graph G = (V G , E G) , a subset S ⊆ V G is called a maximum dissociation set if the induced subgraph G [ S ] does not contain P 3 as its subgraph, and the subset has maximum cardinality. The dissociation number of G is the number of vertices in a maximum dissociation set of G. In this paper, we first characterize all the connected graphs (resp. bipartite graphs, trees) having maximum spectral radius among connected graphs (resp. bipartite graphs, trees) with given order and dissociation number. Secondly, we show that the connected n -vertex graph with dissociation number φ having minimum spectral radius is a tree, where φ ≥ 2 3 n. Finally, we determine the graphs having minimum spectral radius with fixed order n and dissociation number φ ∈ { 2 , ⌈ 2 n / 3 ⌉ , n − 1 , n − 2 }. [ABSTRACT FROM AUTHOR]

The spectral radius and rank of a graph are defined to be the spectral radius and rank of its adjacency matrix, respectively. It is an important problem in spectral extremal graph theory to determine the extremal graph that has the maximum or minimum spectral radius over certain families of graphs. Monsalve and Rada (2021) [8] obtained the extremal graphs with maximum and minimum spectral radii among all graphs with order n and rank 4. In this paper, we first determine the extremal graph which attains the maximum spectral radius among all graphs with any given order n and rank r , and further determine the extremal graph which attains the minimum spectral radius among all graphs with order n and rank 5. [ABSTRACT FROM AUTHOR]

The degree diameter problem is a fundamental and well-studied problem in extremal graph theory, which deals with trade-offs among three parameters: the maximum degree, the diameter, and the number of vertices in a graph. In this paper, we introduce another parameter that represents the robustness of a network and investigate trade-offs among the parameters. For positive integers r and k , we say that a graph is (r , k) -connected if it contains r internally vertex-disjoint paths of length at most k between any pair of vertices. We consider an (r , k) -connected graph with n vertices whose maximum degree is minimized. Our contribution is to show that the minimum of the maximum degree of such a graph is Θ (max { r n k , r }) , which is a tight bound up to constant factors. [ABSTRACT FROM AUTHOR]

Characterizing the graph having the maximum or minimum spectral radius in a given class of graphs is a classical problem in spectral extremal graph theory, originally proposed by Brualdi and Solheid. Given a graph G , a vertex subset S is called a maximum generalized 4-independent set of G if the induced subgraph G [ S ] dose not contain a 4-tree as its subgraph, and the subset S has maximum cardinality. The cardinality of a maximum generalized 4-independent set is called the generalized 4-independence number of G. In this paper, we firstly determine the connected graph (resp. bipartite graph, tree) having the largest spectral radius over all connected graphs (resp. bipartite graphs, trees) with fixed order and generalized 4-independence number, in addition, we establish a lower bound on the generalized 4-independence number of a tree with fixed order. Secondly, we describe the structure of all the n -vertex graphs having the minimum spectral radius with generalized 4-independence number ψ , where ψ ⩾ ⌈ 3 n / 4 ⌉. Finally, we identify all the connected n -vertex graphs with generalized 4-independence number ψ ∈ { 3 , ⌈ 3 n / 4 ⌉ , n − 1 , n − 2 } having the minimum spectral radius. • Characterize the extremal graphs having the maximum spectral radius in some given class of graphs. • Establish a sharp lower bound on the generalized 4-independence number of a tree with fixed order. • Describe the structure of connected graphs in terms of generalized 4-independence numbers and spectral radii. [ABSTRACT FROM AUTHOR]

An interval graph is the intersection graph of a family F of intervals on the real line. An interval order is a partial order (F , ≺) , where F is a family of intervals on the real line, such that for all I 1 , I 2 ∈ F , I 1 ≺ I 2 if and only if I 1 lies entirely to the left of I 2. In both cases, the family F is called a model of the graph (order). The interval count of a given graph (resp. order) is the smallest number of interval lengths needed in any model of this graph (resp. order). The first problem we consider is related to the classes of graphs and orders which can be represented with two interval lengths a and b , with respect to the inclusion hierarchy among such classes for variables a , b. The second problem is an extremal problem which consists of determining the smallest graph or order which has interval count at least k. In particular, we study a conjecture by Fishburn on this extremal problem, verifying its validity when such a conjecture is constrained to the classes of trivially perfect orders and split orders. [ABSTRACT FROM AUTHOR]

For two given measure-preserving functions defined on the unit interval f , g : I → I , the function given by C f , g (u , v) : = λ (f − 1 ([ 0 , u ]) ∩ g − 1 ([ 0 , v ])) is a copula. Although the theoretical problem for constructing this copula is completely solved, in practice it is a rather difficult task. The principal problem is the reverse implication (that is, to prove that f and g are measure-preserving when C f , g is a copula). We provide new proof of this fact with a technique that is far from the previous ones already known in the literature. Indeed, finding two measure-preserving functions f and g , such that C f , g = C , for a given C , is equivalent to a suitable decomposition of such copula in the form C = C f , id ⁎ C id , g (the ⁎ -product), where id denotes the identity function. We also provide explicit algorithms which solve this problem in various contexts such as the measure preserving functions f and g are monotonic, as well as the copula C is a diagonal copula, an extreme copula, an extremal biconic copula, an Archimedean copula, a conic copula, a copula invariant under truncations, or an α -migrative copula. [ABSTRACT FROM AUTHOR]

This paper introduces a new concept of Caputo type interval-valued fractional conformable calculus. Based on this, some theorems and properties related to fractional conformable calculus are presented. As an application, this paper attempts to investigate an initial value problem of functional integro-differential equations with Caputo type interval-valued fractional conformable derivatives. We not only prove the existence of at least one solution, the extremal solutions and the unique solution of the above problem, but also establish some monotone iterative sequences converging to the extremal solutions. More meaningful and important, some comparison principles and generalized Gronwall inequality related to the proposed problem are also proved. Two examples are presented to illustrate the main results. We firmly believe that the concept of interval-valued fractional conformable calculus, new comparison principles and generalized Gronwall inequality introduced here can be conveniently applied to qualitative analysis of a variety of new interval-valued fractional conformable biological models, economic models, etc., which will better describe and explain various complex phenomena derived from the real world. [ABSTRACT FROM AUTHOR]

The [ n , k , r ] -Locally recoverable codes (LRC) studied in this work are a well-studied family of [ n , k ] linear codes for which the value of each symbol can be recovered by a linear combination of at most r other symbols. In this paper, we study the LMD problem, which is to find the largest possible minimum distance of [ n , k , r ] -LRCs, denoted by D (n , k , r). LMD can be approximated within an additive term of one—it is known that D (n , k , r) is equal to either d ⁎ or d ⁎ − 1 , where d ⁎ = n − k − ⌈ k r ⌉ + 2. Moreover, for a range of parameters, it is known precisely whether the distance D (n , k , r) is d ⁎ or d ⁎ − 1. However, the problem is still open despite a significant effort. In this work, we convert LMD to an equivalent simply-stated problem in graph theory. Using this conversion, we show that an instance of LMD is at least as hard as computing the size of a maximal graph of high girth, a hard problem in extremal graph theory. This is an evidence that LMD—although can be approximated within an additive term of one—is hard to solve in general. As a positive result, thanks to the conversion and the exiting results in extremal graph theory, we solve LMD for a range of code parameters that has not been solved before. [ABSTRACT FROM AUTHOR]

A graph is minimally k -(edge)-connected if it is k -connected (respectively, k -edge-connected) and deleting any arbitrary chosen edge always leaves a graph which is not k -connected (respectively, k -edge-connected). What is the maximum (signless Laplacian) spectral radius and what are the corresponding extremal graphs among minimally k -(edge)-connected graphs for k ≥ 2 ? Chen and Guo (2019) gave the answer to k = 2 and characterized the corresponding extremal graphs. In this paper, we first give the answer to k = 3 for minimally 3-connected graphs. For the signless Laplacian spectral radius, we also consider the problem for k = 2 , 3 and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

Let C ℓ be the cycle of order ℓ. The square of C ℓ , denoted by C ℓ 2 , is obtained by joining all pairs of vertices with distance no more than two in C ℓ. Denote by ex (n , F) and spex (n , F) the maximum size and maximum spectral radius over all n -vertex F -free graphs, respectively. The well-known Turán problem asks for ex (n , F) , and Nikiforov in 2010 proposed a spectral counterpart, known as the Brualdi-Solheid-Turán type problem, focusing on determining spex (n , F). In this paper, for any integer ℓ ≥ 6 that is not divisible by 3, we characterize the unique extremal graph with respect to ex (n , C ℓ 2) and spex (n , C ℓ 2) for sufficiently large n , respectively. [ABSTRACT FROM AUTHOR]

The total number of independent subsets, including the empty set, of a graph, is also termed as the Merrifield–Simmons index (MSI) in mathematical chemistry. The connective eccentricity index (CEI), the multiplicative Wiener index (MWI), and the Wiener polarity index (WPI) are all distance-based topological indices. Some of these topological indices found applications in chemistry. The eccentric complexity of a graph G is defined to be the number of different eccentricities of its vertices. In this paper, we first investigate the relationship between MSI and three other distance-based topological indices (CEI, MWI, WPI). We prove that MSI > WPI for any tree, MSI > CEI for any connected bipartite graph, and MWI > MSI for connected graphs with given restricted condition on size. Then we consider extremal problems of MSI on graphs with small eccentric complexity. Moreover, we prove an inequality relating MSI, independence number and eccentric complexity for some special graph families. Finally, some further problems and conjectures are proposed. [ABSTRACT FROM AUTHOR]

A spectral version of extremal graph theory problem is as follows: what is the maximum spectral radius of an H -free graph of order n ? Nikiforov posed a conjecture on spectral extremal value of cycles. In the past decade, much attention has been paid to this conjecture and other questions on spectral extremal graphs. In order to approach to Nikiforov's conjecture, we provide a new conjecture on spectral extremal value involving fans and wheels. Moreover, we show it is true for k = 2. [ABSTRACT FROM AUTHOR]

In this paper, the extremal problems on the phenylene chains with respect to some graph invariants are studied. All the graphs minimizing (resp. maximizing) the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index among all the phenylene chains each of which contains n four-membered rings are identified. [ABSTRACT FROM AUTHOR]

Abstract Let Ω be a domain of R n , n ≥ 3. The classical Caffarelli–Kohn–Nirenberg inequality rewrites as the following inequality: for any s ∈ [ 0 , 2 ] and any γ < (n − 2) 2 4 , there exists a constant K (Ω , γ , s) > 0 such that (H S) ∫ Ω | u | 2 ⋆ (s) | x | s d x 2 2 ⋆ (s) ≤ K (Ω , γ , s) ∫ Ω | ∇ u | 2 − γ u 2 | x | 2 d x , for all u ∈ D 1 , 2 (Ω) (the completion of C c ∞ (Ω) for the relevant norm). When 0 ∈ Ω is an interior point, the range (− ∞ , (n − 2) 2 4) for γ cannot be improved: moreover, the optimal constant K (Ω , γ , s) is independent of Ω and there is no extremal for (H S). But when 0 ∈ ∂ Ω , the situation turns out to be drastically different since the geometry of the domain impacts : • the range of γ 's for which (H S) holds. • the value of the optimal constant K (Ω , γ , s) ; • the existence of extremals for (H S). When Ω is smooth, the problem was tackled by Ghoussoub–Robert (2017) where the role of the mean curvature was central. In the present paper, we consider nonsmooth domain with a singularity at 0 modeled on a cone. We show how the local geometry induced by the cone around the singularity influences the value of the Hardy constant on Ω. When γ is small, we introduce a new geometric object at the conical singularity that generalizes the "mean curvature": this allows to get extremals for (H S). The case of larger values for γ will be dealt in the forthcoming paper (Cheikh-Ali, 2018). As an intermediate result, we prove the symmetry of some solutions to singular pdes that has an interest on its own. [ABSTRACT FROM AUTHOR]

Abstract Let C ℓ be a cycle of length ℓ and S n , k = K k ∨ K n − k ‾ , the join graph of a complete graph of order k and an empty graph on n − k vertices, and S n , k + be the graph obtained from S n , k by adding an edge in the independent set of S n , k. Nikiforov conjectured that for a given integer k ≥ 2 , any graph G of sufficiently large order n with spectral radius μ (G) ≥ μ (S n , k) (or μ (G) ≥ μ (S n , k +)) contains C 2 k + 1 or C 2 k + 2 (or C 2 k + 2), unless G = S n , k (or G = S n , k +). In this paper, a weaker version of Nikiforov's conjecture is considered, we prove that for a given integer k ≥ 2 , any graph G of sufficiently large order n with spectral radius μ (G) ≥ μ (S n , k) (or μ (G) ≥ μ (S n , k +)) contains a cycle C ℓ with ℓ ≥ 2 k + 1 (or C ℓ with ℓ ≥ 2 k + 2), unless G = S n , k (or G = S n , k +). These results also imply a result given by Nikiforov in (2010) [3, Theorem 2]. [ABSTRACT FROM AUTHOR]

Abstract We consider an initial–boundary value problem for a quasilinear parabolic system of hemivariational inequalities which is not necessarily coercive. The system exhibits full dependence on the gradient of the solution and is doubly coupled on both the source and multivalued terms. Based on sub-supersolutions, truncation functions, and nonsmooth analysis we establish an existence and enclosure result for weak solutions within a trapping region. The existence of extremal solutions is proved without imposing any monotonicity conditions on lower order terms. Moreover, we present a sufficient condition to construct effectively a trapping region and show the existence of positive solutions. [ABSTRACT FROM AUTHOR]

Let F denote a set of graphs. A graph G is said to be F -free if it does not contain any element of F as a subgraph. The Turán number is the maximum possible number of edges in an F -free graph with n vertices. It is well known that classical Turán type extremal problem aims to study the Turán number of fixed graphs. In 2010, Nikiforov [17] proposed analogously a spectral Turán type problem which asks to determine the maximum spectral radius of an F -free graph with n vertices. It attracts much attention and many such problems remained elusive open even after serious attempts, and so they are considered as one of the most intriguing problems in spectral extremal graph theory. It is interesting to consider another spectral Turán type problem which asks to determine the maximum spectral radius of an F -free graph with m edges. Denote by G (m , F) the set of F -free graphs with m edges having no isolated vertices. Each of the graphs among G (m , F) having the largest spectral radius is called a maximal graph. Let θ p , q , r be a theta graph formed by connecting two distinct vertices with three independent paths of length p , q and r , respectively (length refers to the number of edges). In this paper, we firstly determine the unique maximal graph among G (m , θ 1 , 2 , 3) and G (m , θ 1 , 2 , 4) , respectively. Then we determine all the maximal graphs among G (m , C 5) (resp. G (m , C 6)) excluding the book graph. These results extend some earlier results. [ABSTRACT FROM AUTHOR]

Abstract It is known that a positive Boolean function f depending on n variables has at least n + 1 extremal points, i.e. minimal ones and maximal zeros. We show that f has exactly n + 1 extremal points if and only if it is linear read-once. The class of linear read-once functions is known to be the intersection of the classes of read-once and threshold functions. Generalizing this result we show that the class of linear read-once functions is the intersection of read-once and Chow functions. We also find the set of minimal read-once functions which are not linear read-once and the set of minimal threshold functions which are not linear read-once. In other words, we characterize the class of linear read-once functions by means of minimal forbidden subfunctions within the universe of read-once and the universe of threshold functions. Within the universe of threshold functions the importance of linear read-once functions is due to the fact that they attain the minimum value of the specification number, which is n + 1 for functions depending on n variables. In 1995 Anthony et al. conjectured that for all other threshold functions the specification number is strictly greater than n + 1. We disprove this conjecture by exhibiting a threshold non-linear read-once function depending on n variables whose specification number is n + 1. [ABSTRACT FROM AUTHOR]

Abstract Strong extrema of functions on some quasi-metric and compact spaces are studied. It is shown that the maximum cardinalities of the set of strong extrema of discontinuous functions can be larger than for continuous ones. We construct quasi-metric spaces with uncountable weight, on which any function has no more than a countable set of strong extrema. [ABSTRACT FROM AUTHOR]

Let G be a graph with adjacency matrix A ( G ) and let D ( G ) be the diagonal matrix of the degrees of G . For any real α ∈ [ 0 , 1 ] , Nikiforov (2017) [7] defined the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . Let u and v be two vertices of a connected graph G . Suppose that u and v are connected by a path w 0 ( = v ) w 1 ⋯ w s − 1 w s ( = u ) where d ( w i ) = 2 for 1 ≤ i ≤ s − 1 . Let G p , s , q ( u , v ) be the graph obtained by attaching the paths P p to u and P q to v . Let s = 0 , 1 . Nikiforov and Rojo (2018) [9] conjectured that ρ α ( G p , s , q ( u , v ) ) < ρ α ( G p − 1 , s , q + 1 ( u , v ) ) if p ≥ q + 2 . In this paper, we confirm the conjecture. As applications, firstly, the extremal graph with maximal A α -spectral radius with fixed order and cut vertices is characterized. Secondly, we characterize the extremal tree which attains the maximal A α -spectral radius with fixed order and matching number. These results generalize some known results. [ABSTRACT FROM AUTHOR]

Let C v ( k ; T ) be the number of closed walks of length k starting at vertex v in a tree T . We prove that for any tree T with a given degree sequence π , the vector C ( k ; T ) ≡ ( C v ( k ; T ), v  ∈  V ( T )) is weakly majorized by the vector C ( k ; T π * ) ≡ ( C v ( k ; T π * ) , v ∈ V ( T π * ) ) , where T π * is the greedy tree with the degree sequence π . In addition, for two trees degree sequences π and π ′, if π is majorized by π ′, then C ( k ; T π * ) is weakly majorized by C ( k ; T π ′ * ) . [ABSTRACT FROM AUTHOR]

Abstract Addressing spatial variation of extreme precipitation in urban areas is important for urban hydrologic designs. Climatology of urban areas is, in general, different from that of its surroundings and the spatial variation of extreme precipitation within the city exhibits shorter spatial range, especially for short duration events. This work aims at understanding the dependence structure of extreme precipitation within an urban area and its surrounding non-urban areas at various durations. The spatial dependence of precipitation is analysed with three different measures for examining dependence, considering observations from pairs of stations. Further, the spatial precipitation extremes are modelled with the max-stable process to include the dependence structure of spatial extremes. The City of Berlin, Germany, with surrounding non-urban area is considered to demonstrate the methodology. For this case study, the hourly precipitation shows weaker dependence within the city even at small distances, whereas the daily precipitation shows a high degree of dependence. This dependence structure of the daily precipitation gets masked as more and more surrounding non-urban areas are included in the analysis. Further, the extreme precipitation at different durations are modelled considering max-stable process. Different geographical and climatological covariates are considered in the modelling of location and scale parameters of the Generalized Extreme Value distribution. The geographical covariates are seen to be predominant within the city and the climatological covariates prevail when non-urban areas are added. These results suggest the importance of quantification of dependence structure of spatial precipitation at the sub-daily timescales, as well as the need to more precisely model spatial extremes within the urban areas. [ABSTRACT FROM AUTHOR]

An extremal Kähler metric with finite singularities on a compact Riemann surface is often called an HCMU metric. It can be regarded as a generalization of CSC (constant scalar curvature) metric. In this paper we will prove that any HCMU metric is a pullback of an HCMU metric on S 2 with only two singularities (we call it football) by a multi-valued holomorphic function. Then we study the monodromy group of the multi-valued function. [ABSTRACT FROM AUTHOR]

A connected graph is said to be a cactus if each of its blocks is either a cycle or an edge. Let C n be the set of all n -vertex cacti with circumference at least 4, and let C n , k be the set of all n -vertex cacti containing exactly k ⩾ 1 cycles where n ⩾ 3 k + 1 . In this paper, lower bounds on the difference between the (revised) Szeged index and Wiener index of graphs in C n (resp. C n , k ) are proved. The minimum and the second minimum values on the difference between the Szeged index and Wiener index of graphs among C n are determined. The bound on the minimum value is strengthened in the bipartite case. A lower bound on the difference between the revised Szeged index and Wiener index of graphs among C n , k is also established. Along the way the corresponding extremal graphs are identified. [ABSTRACT FROM AUTHOR]

This paper considers the degree-diameter problem for undirected circulant graphs. For degrees 10 and 11, newly discovered families of circulant graphs of arbitrary diameter are presented which are largest known and are conjectured to be extremal. They are also the largest-known Abelian Cayley graphs of these degrees. For each such family, the order of every graph in the family is defined by a quintic polynomial function of the diameter which is specific to the family. The elements of the generating set for each graph are similarly defined by a set of polynomials in the diameter. The existence of the graphs in the degree 10 families has been proved for all diameters. These graphs are consistent with a conjecture on the order of extremal Abelian Cayley and circulant graphs of any degree and diameter. [ABSTRACT FROM AUTHOR]

For a graph G = ( V , E ) , the eccentric connectivity index of G , denoted by ξ c ( G ), is defined as ξ c ( G ) = ∑ v ∈ V ɛ ( v ) d ( v ) , where ɛ( v ) and d ( v ) are the eccentricity and the degree of v in G , respectively. In this paper, we first establish the sharp lower bound for the eccentric connectivity index in terms of the order and the minimum degree of a connected G , and characterize some extremal graphs, which generalize some known results. Secondly, we characterize the extremal trees having the maximum or minimum eccentric connectivity index for trees of order n with given degree sequence. Finally, we give a sharp lower bound for the eccentric connectivity index in terms of the order and the radius of a unicyclic G , and characterize all extremal graphs. [ABSTRACT FROM AUTHOR]

The generalized A B C index of a graph G , denoted by A B C α ( G ) , is defined as the sum of weights ( d i + d j − 2 d i d j ) α over all edges v i v j of G , where α is an arbitrary non-zero real number, and d i is the degree of vertex v i of G . In this paper, we first prove that the generalized A B C index of a connected graph will increase with addition of edge(s) if α < 0 or 0 < α ≤ 1 ∕ 2 , which provides a useful tool for the study of extremal properties of the generalized A B C index. By means of this result, we then characterize the graphs having the maximal A B C α value for α < 0 among all connected graphs with given order and vertex connectivity, edge connectivity, or matching number. Our work extends some previously known results. [ABSTRACT FROM AUTHOR]

For two graphs G and F , the extremal number of F in G , denoted by ex (G , F) , is the maximum number of edges in a spanning subgraph of G not containing F as a subgraph. Determining ex (K n , F) for a given graph F is a classical extremal problem in graph theory. In 1962, Erdős determined ex (K n , k K 3) , which generalized Mantel's Theorem. On the other hand, in 1974, Bollobás, Erdős, and Straus determined ex (K n 1 , n 2 , ... , n r , K t) , which extended Turán's Theorem to complete multipartite graphs. In this paper, we determine ex (K n 1 , n 2 , ... , n r , k K 3) for r ≥ 4 and 10 k − 4 ≤ n 1 + 4 k ≤ n 2 ≤ n 3 ≤ ⋯ ≤ n r. [ABSTRACT FROM AUTHOR]

The revised Szeged index of a graph is defined as S z * ( G ) = ∑ e = u v ∈ E ( n u ( e ) + n 0 ( e ) 2 ) ( n v ( e ) + n 0 ( e ) 2 ) , where n u ( e ) and n v ( e ) are, respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u , and n 0 ( e ) is the number of vertices equidistant to u and v . In the paper, we identify the lower bound of revised Szeged index among all bicyclic graphs, and also characterize the extremal graphs that attain the lower bound. [ABSTRACT FROM AUTHOR]

We prove weak convergence on the Skorokhod space of Galton–Watson processes with immigration, properly normalized, under the assumption that the tail of the immigration distribution has a logarithmic decay. The limits are extremal shot noise processes. By considering marginal distributions, we recover the results of Pakes (1979). [ABSTRACT FROM AUTHOR]

The Wiener index W ( G ) of a connected graph G is the sum of distances between all pairs of vertices of G . A connected graph G is said to be a cactus if each of its blocks is either a cycle or an edge. Let G n , t be the set of all n -vertex cacti containing exactly t cycles. Liu and Lu (2007) determined the unique graph in G n , t with the minimum Wiener index. We now establish a sharp upper bound on the Wiener index of graphs in G n , t and identify the corresponding extremal graphs. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded smooth domain in R n , W 1 , n ( Ω ) be the Sobolev space on Ω , and λ ( Ω ) = inf { ‖∇u‖ n n : ∫ Ω u d x = 0 , ‖ u ‖ n = 1 } be the first nonzero Neumann eigenvalue of the n − Laplace operator − Δ n on Ω . For 0 ≤ α < λ ( Ω ) , let us define ‖u‖ 1 , α n = ‖∇u‖ n n − α ‖u‖ n n . We prove, in this paper, the following improved Moser–Trudinger inequality on functions with mean value zero on Ω , sup u ∈ W 1 , n ( Ω ) , ∫ Ω u d x = 0 , ‖ u ‖ 1 , α = 1 ∫ Ω e β n | u | n n − 1 d x < ∞ , where β n = n ( ω n − 1 ∕ 2 ) 1 ∕ ( n − 1 ) , and ω n − 1 denotes the surface area of unit sphere in R n . We also show that this supremum is attained by some function u ∗ ∈ W 1 , n ( Ω ) such that ∫ Ω u ∗ d x = 0 and ‖ u ∗ ‖ 1 , α = 1 . This generalizes a result of Ngo and Nguyen (0000) in dimension two and a result of Yang (2007) for α = 0 , and improves a result of Cianchi (2005). [ABSTRACT FROM AUTHOR]

Recent measurements of the proton–proton total cross section σ t o t at 7 and 8 TeV by the TOTEM and ATLAS Collaborations are characterized by some discrepant values: the TOTEM data suggest a rise of the cross section with the energy faster than the ATLAS data. Attempting to quantify these different behaviors, we develop new analytical fits to σ t o t and ρ data from pp and p ¯ p scattering in the energy region 5 GeV–8 TeV. The dataset comprises all the accelerator data below 7 TeV and we consider three ensembles by adding: either only the TOTEM data (T), or only the ATLAS data (A), or both sets (T + A). For the purposes, we use our previous RRPL γ parametrization for σ t o t ( s ) , consisting of two Reggeons (RR), one critical Pomeron (P) and a leading log-raised-to-gamma (L γ ) contribution (with γ as a free fit parameter), analytically connected to ρ ( s ) through singly-subtracted derivative dispersion relations and energy scale fixed at the physical threshold. The data reductions with ensembles T and A present good agreement with the experimental data analyzed and cannot be distinguished on statistical grounds. The quality of the fit is not as good with ensemble T + A. The fit results provide γ ∼ 2.3 ± 0.1 (T), 2.0 ± 0.1 (A), 2.2 ± 0.2 ( T + A ), with χ 2 / DOF ∼ 1.07 (T), 1.09 (A), 1.14 (T + A), suggesting extrema bounds for γ given by 1.9 and 2.4. Fits with γ = 2 (fixed) are also developed and discussed. [ABSTRACT FROM AUTHOR]

In this note we explore the extremal graphs with given matching number with respect to topological indices. We present generalization of previous studies on such graphs and apply our findings to various indices that have not yet been considered for graphs with given matching number. [ABSTRACT FROM AUTHOR]

The general Randić index of a graph G is defined as R α ( G ) = ∑ u v ∈ E ( G ) ( d ( u ) d ( v ) ) α , where d ( u ) denotes the degree of a vertex u in G and α is a real number. In this paper, we determine the maximum general Randić indices of trees and chemical trees with n vertices and k pendent vertices for 4 ≤ k ≤ ⌊ n + 2 3 ⌋ and α 0 ≤ α < 0, where α 0 ≈ − 0.5122 is the unique non-zero root of the equation 6 · 4 α − 20 · 9 α + 10 · 12 α − 16 α + 5 · 24 α = 0 . The corresponding extremal graphs are also characterized. [ABSTRACT FROM AUTHOR]

We investigate the strategies used by 64 advanced secondary mathematics students to identify whether a given pair of polynomial representations (graphs, tables, or equations) corresponded to the same function on an assessment of coordinating representations. Participants also completed assessments of domain-related knowledge and background skills. Cluster analysis of strategies by representation pair revealed patterns in the participants' strategy use. Two clusters were identifiable on tasks that required matching equations to graphs and graphs to tables. We identified overlap between these two clusters, suggesting that while the representation pair influenced strategy choice, there was also a general distinction between students who used more and less sophisticated strategies. However, students who used more sophisticated coordination strategies were similar to the others on measures of domain-specific knowledge or background skills. We consider implications for future investigations testing interventions to promote coordinating representations. [ABSTRACT FROM AUTHOR]

A segment of a tree T is a path whose end vertices have degree 1 or at least 3, while all internal vertices have degree 2. The lengths of all the segments of T form its segment sequence, in analogy to the degree sequence. We address a number of extremal problems for the class of all trees with a given segment sequence. In particular, we determine the extremal trees for the number of subtrees, the number of matchings and independent sets, the graph energy, and spectral moments. [ABSTRACT FROM AUTHOR]

The atom-bond connectivity ( A B C ) index of a graph G is defined as the sum of the weights d ( u ) + d ( v ) − 2 d ( u ) d ( v ) of all edges u v of G , where d ( u ) denotes the degree of a vertex u in G . In this short note, we determine the maximum A B C index for graphs with edge-connectivity one and characterize the corresponding extremal graphs. This answers a recent question proposed by Zhang et al. (2016). [ABSTRACT FROM AUTHOR]

Given a family of graphs F , a graph G is F -saturated if G contains no member of F as a subgraph, but G + e contains a member of F as a subgraph for each edge e in the complement of G. The edge spectrum of F is the set of all possible sizes of F -saturated graphs on n vertices. A G -subdivision is a graph derived from G by replacing each edge of G with a path of arbitrary length. Let C ≥ k denote the family of C k -subdivisions, where C k is a cycle of length k with k ≥ 3. Determining the minimum or maximum number of edges in n -vertex F -saturated graphs are two of the most important problems in the study of extremal graph theory. This is also a very important optimization problem in graph theory. The study of this problem is closely related to the development of other branches of mathematics, computer science, network, modern information science and technology. In this paper, we determine the edge spectrum of C ≥ k for each k ∈ { 3 , 4 , 5 , 6 }. [ABSTRACT FROM AUTHOR]

A signed graph Σ = (G , σ) consists of a underlying graph G = (V , E) with a signature function σ : E → { − 1 , 1 }. ρ (M) denotes the M -spectral radius of Σ, where M = M (Σ) is a real symmetric graph matrix of Σ. Obviously, ρ (M) = max { λ 1 (M) , − λ n (M) }. Let A (Σ) be the adjacency matrix of Σ and (K n , H −) be a signed complete graph whose negative edges induce a subgraph H. In this paper, we first focus on a central problem in spectral extremal graph theory: which spanning tree T makes the ρ (A (Σ)) of the unbalanced (K n , T −) as large as possible? To answer the problem, we characterize the extremal signed graph with maximum λ 1 (A (Σ)) and minimum λ n (A (Σ)) among graphs of type (K n , T −) , respectively. Another matrix associated to a signed graph is the distance matrix D (Σ) defined by Hameed, Shijin, Soorya, Germina and Zaslavsky. Note that A (Σ) = D (Σ) when Σ ∈ (K n , T −). In this paper, we give upper bounds on the least distance eigenvalue of a signed graph Σ with diameter at least 2. This result implies a result proved by Lin originally conjectured by Aouchiche and Hansen. [ABSTRACT FROM AUTHOR]

A cactus is a connected graph in which any two cycles have at most one common vertex. The distance spectral radius ρ ( G ) of a graph G is the largest eigenvalue of the distance matrix D ( G ). Recently, many researchers proposed the use of ρ ( G ) as a molecular structure descriptor of alkanes. In this paper, we characterize n -vertex cyclic cactus with given matching number m which minimizes the distance spectral radius. The resulting cactus also minimizes the Hosoya index, the Wiener index and the Randić index in the same class of graphs. [ABSTRACT FROM AUTHOR]

Two efficient image fusion algorithms are proposed for constructing a fused image through combining parallel features on multi-scale local extrema scheme. Firstly, the source image is decomposed into a series of smoothed and detailed images at different scales by local extrema scheme. Secondly, the parallel features of edge and color are extracted to get the saliency maps. The edge saliency weighted map aims to preserve the structural information using Canny edge detection operator; Meanwhile, the color saliency weighted map works for extracting the color and luminance information by context-aware operator. Thirdly, the average and weighted average schemes are used as the fusion rules for grouping the coefficients of weighted maps obtained from smoothed and detailed images. Finally, the fused image is reconstructed by the fused smoothed and the fused detailed images. Experimental results demonstrate that the proposed algorithms show the best performances among the other fusion methods in the domain of MRI-CT and MRI-PET fusion. [ABSTRACT FROM AUTHOR]

We show that every C 6 -free graph G has a C 4 -free, bipartite subgraph with at least 3 e ( G ) / 8 edges. Our proof is probabilistic and uses a theorem of Füredi et al. (2006) on C 6 -free graphs. [ABSTRACT FROM AUTHOR]

Let S = ( a 1 , … , a m ; b 1 , … , b n ) , where a 1 , … , a m and b 1 , … , b n are two nonincreasing sequences of nonnegative integers. The pair S = ( a 1 , … , a m ; b 1 , … , b n ) is said to be a bigraphic pair if there is a simple bipartite graph G = ( X ∪ Y , E ) such that a 1 , … , a m and b 1 , … , b n are the degrees of the vertices in X and Y , respectively. Let A be an (additive) Abelian group. We define σ ( A , m , n ) to be the minimum integer k such that every bigraphic pair S = ( a 1 , … , a m ; b 1 , … , b n ) with a m , b n ≥ 2 and σ ( S ) = a 1 + ⋯ + a m ≥ k has an A -connected realization. In this paper, we determine the values of σ ( Z 3 , m , m ) for m ≥ 4 . [ABSTRACT FROM AUTHOR]

Let G = ( V , E ) be a connected graph with vertex set V ( G ) = { v 1 , v 2 , … , v n } and edge set E ( G ) . The transmission Tr ( v i ) of vertex v i is defined to be the sum of distances from v i to all other vertices. Let Tr ( G ) be the n × n diagonal matrix with its ( i , i ) -entry equal to Tr G ( v i ) . The distance signless Laplacian is defined as D Q ( G ) = Tr ( G ) + D ( G ) , where D ( G ) is the distance matrix of G . Let ∂ 1 ( G ) ≥ ∂ 2 ( G ) ≥ ⋯ ≥ ∂ n ( G ) denote the eigenvalues of distance signless Laplacian matrix of G . In this paper, we first characterize all graphs with ∂ n ( G ) = n − 2 . Secondly, we characterize all graphs with ∂ 2 ( G ) ∈ [ n − 2 , n ] when n ≥ 11 . Furthermore, we give the lower bound on ∂ 2 ( G ) with independence number α and the extremal graph is also characterized. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of two extremal problems arising naturally in heat conduction processes. We look for optimal configurations of thermal axisymmetric fins and model this problem as the issue of (i) minimizing (for the worst shape) or (ii) maximizing (for the best shape) the first eigenvalue of a selfadjoint operator having a compact inverse. We impose a pointwise lower bound on the radius of the fin, as well as a lateral surface constraint. Using particular perturbations and under a smallness assumption on the pointwise lower bound, one shows that the only solution is the cylinder in the first case whereas there is no solution in the second case. We moreover construct a maximizing sequence and provide the optimal value of the eigenvalue in this case. As a byproduct of this result, and to propose a remedy to the non-existence in the second case, we also investigate the well-posedness character of another optimal design problem set in a class enjoying good compactness properties. [ABSTRACT FROM AUTHOR]

The atom-bond connectivity (ABC) index is a degree-based graph topological index that found chemical applications, including those of predicting the stability of alkanes and the strain energy of cycloalkanes. Several structural properties of the trees with minimal ABC index were proved recently, however the complete characterization of the minimal-ABC trees is still an open problem. It is known that minimal-ABC trees can have at most one pendent path of length 3. It is also known that the minimal-ABC trees that have a pendent path of length 3 do not contain so-called B k -branches, with k ≥ 4, and do not contain more than two B 2 -branches. Here, we improve the latter result by showing that minimal-ABC trees of order larger than 168 and with a pendent path of length 3 do not contain B 2 -branches. Moreover, we show that trees with minimal ABC index with a pendent path of length 3 do not contain B 1 -branches. [ABSTRACT FROM AUTHOR]

The Krein–Milman theorem states that every compact convex subset in a locally compact convex space is the closure of the convex hull of its extreme points. Inspired by this result, we investigate the existence of extreme points in compact convex subsets of asymmetric normed spaces. We focus our attention in the finite dimensional case, giving a geometric description of all compact convex subsets of a finite dimensional asymmetric normed space. [ABSTRACT FROM AUTHOR]