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In the paper, we introduce the signless Laplacian A B C -matrix Q ̃ (G) = D ¯ (G) + A ̃ (G) , where D ¯ (G) is the diagonal matrix of A B C -degrees and A ̃ (G) is the A B C -matrix of G. The eigenvalues of the matrix Q ̃ (G) are the signless Laplacian A B C -eigenvalues of G. We give some basic properties of the matrix Q ̃ (G) , which includes relating independence number and clique number with signless Laplacian A B C -eigenvalues. For bipartite graphs, we show that the signless Laplacian A B C -spectrum and the Laplacian A B C -spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian A B C -eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian A B C -eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix Q ̃ (G) − t r (Q ̃ (G)) n I , called the signless Laplacian A B C -energy of G. We obtain some upper and lower bounds for signless Laplacian A B C -energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the A B C -energy with the signless Laplacian A B C -energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian A B C -energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same A B C -energy. [ABSTRACT FROM AUTHOR]

The power graph P (Z n) of Z n for a finite cyclic group Z n is a simple undirected connected graph such that two distinct nodes x and y in Z n are adjacent in P (Z n) if and only if x ≠ y and x i = y or y i = x for some non-negative integer i . In this article, we find the Laplacian eigenvalues of P (Z n) and show that P (Z n) is Laplacian integral (integer algebraic connectivity) if and only if n is either the product of two distinct primes or a prime power. That answers a conjecture by Panda, Graphs and Combinatorics, (2019). [ABSTRACT FROM AUTHOR]

Graph clustering aims to divide nodes into different clusters without labels and has attracted great attention due to the success of graph neural networks (GNNs). Traditional GNN-based clustering methods are based on the homophilic assumption, i.e., connected nodes belong to the same clusters. However, this assumption is not always true, as heterophilic graphs are also ubiquitous in the real world, which limits the application of GNNs. Furthermore, these methods overlook global positions, which can result in erroneous clustering. To solve the aforementioned problems, we propose a novel model called Preserving Global Information for Graph Clustering with Masked Autoencoders (GCMA). We first propose a low–high-pass filter to capture meaningful low- and high-frequency information. Then, we propose a graph diffusion method to obtain the global position. Specifically, a parameterized Laplacian matrix is proposed to better control the global direction. To further enhance the learning ability of the autoencoders, we design a model with a masking strategy that enhances the learning ability. Extensive experiments on both homophilic and heterophilic graphs demonstrate GCMA's advantages over state-of-the-art baselines. [ABSTRACT FROM AUTHOR]

The use of the global mean first-passage time (GMFPT) in random walks on networks has been widely explored in the field of statistical physics, both in theory and practical applications. The GMFPT is the estimated interval of time needed to reach a state j in a system from a starting state i. In contrast, there exists an intrinsic measure for a stochastic process, known as Kemeny's constant, which is independent of the initial state. In the literature, it has been used as a measure of network efficiency. This article deals with a graph-spectrum-based method for finding both the GMFPT and Kemeny's constant of random walks on spiro-ring networks (that are organic compounds with a particular graph structure). Furthermore, we calculate the Laplacian matrix for some specific spiro-ring networks using the decomposition theorem of Laplacian polynomials. Moreover, using the coefficients and roots of the resulting matrices, we establish some formulae for both GMFPT and Kemeny's constant in these spiro-ring networks. [ABSTRACT FROM AUTHOR]

Many properties of the structure and dynamics of complex networks derive from the characteristics of the spectrum of the associated Laplacian matrix, specifically from the set of its eigenvalues. In this paper, we show that there exist graphs for which the ratio between the length of the spectrum (that is, the difference between the largest and smallest eigenvalues of the Laplacian matrix) and its spread (the difference between the second smallest eigenvalue and the smallest one) is equal to the golden ratio. We call such graphs Golden Laplacian Graphs (GLG). In this paper, we first find all such graphs with a number of nodes n ≤ 10 . We then prove several graph-theoretic and algebraic properties that characterize these graphs. These graphs prove to be extremely robust, as they have large vertex and edge connectivity along with a large isoperimetric constant. Finally, we study the synchronization properties of GLGs, showing that they are among the top synchronizable graphs of the same size. Therefore, GLGs represent very good candidates for engineering and communication networks. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of Harary Laplacian-energy-like for a simple undirected and connected graph G with order n. We also establish novel matrix results in this regard. Furthermore, by employing matrix order reduction techniques, we derive upper and lower bounds utilizing existing graph invariants and vertex connectivity. Finally, we characterize the graphs that achieve the aforementioned bounds by considering the generalized join operation of graphs. [ABSTRACT FROM AUTHOR]

Using the notions of clique partitions and edge clique covers of graphs, we consider the corresponding incidence structures. This connection furnishes lower bounds on the negative eigenvalues and their multiplicities associated with the adjacency matrix, bounds on the incidence energy, and on the signless Laplacian energy for graphs. For the more general and well-studied set S (G) of all real symmetric matrices associated with a graph G, we apply an extended version of an incidence matrix tied to an edge clique cover to establish several classes of graphs that allow two distinct eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper, we bring forward a distributed algorithm for the assignment of a prescribed Laplacian spectrum for path graphs by means of asymmetric weight assignment. We first describe the meaningfulness and the relevance of this mathematical setting in modern technological applications, and some examples are reported, revealing its practical usefulness in a variety of applications. Then, the solution is derived both theoretically and through an algorithm. Special attention is devoted to a distributed implementation of the main algorithm, which is a valuable feature for several modern applications. Finally, the positivity is discussed, which is revealed to be a consequence of the strict interlacing property. [ABSTRACT FROM AUTHOR]

This brief paper shows that robust chaotic behavior can be detected within a tuned-collector single-stage transistor-based oscillator. The content of this work also contributes to the problem of chaos localization in simplified mathematical model of standard analog building block. Searching for chaos is performed via numerical optimization routine applied onto the principal schematic of oscillator where generalized bipolar transistor is modelled as a two-port described by impedance as well as admittance matrix. In both cases, the presence of dense chaotic attractor is proved via calculation of the largest Lyapunov exponent, while its structural stability is validated by real measurement, i.e., visualization of captured oscilloscope screenshots. [ABSTRACT FROM AUTHOR]

Delay and especially delay in the transmission of agents' information, is one of the most important causes of disruption to achieving consensus in a multi-agent system. This paper deals with achieving consensus in delayed fractional-order multi-agent systems (FOMAS). The aim in the present note is to find the exact maximum allowable delay in a FOMAS with non-uniform delay, i.e., the case in which the interactions between agents are subject to non-identical communication time-delays. By proving a stability theorem, the results available for non-delayed networked fractional-order systems are extended for the case in which interaction links have nonequal communication time-delays. In this extension by considering a time-delay coordination algorithm, necessary and sufficient conditions on the time delays and interaction graph are presented to guarantee the coordination. In addition, the delay-dependent stability region is also obtained. Finally, the dependency of the maximum allowable delay on two parameters, the agent fractional-order and the largest eigenvalue of the graph Laplacian matrix, is exactly determined. Numerical simulation results are given to confirm the proposed methodologies. [ABSTRACT FROM AUTHOR]

This paper studies the consensus of first-order discrete-time multi-agent systems with fixed and switching topology, and there exists cooperative and antagonistic interactions among agents. A signed graph is used to model the interactions among agents, and some sufficient conditions for consensus are obtained by analyzing the eigenvalues of a Laplacian matrix in the case of fixed topology. The results indicate that having a spanning tree is only a necessary condition for the consensus of multi-agent systems with signed graphs, which is also affected by edge weights. Consensus is further discussed in the case of switching topology, and the results reveal that consensus can be reached if the controller gain and the union graphs among some consecutive time intervals satisfy some conditions. Finally, several simulation examples further confirm the theoretical results. [ABSTRACT FROM AUTHOR]

The target of the multi-hop knowledge base question-answering task is to find answers of some factoid questions by reasoning across multiple knowledge triples in the knowledge base. Most of the existing methods for multi-hop knowledge base question answering based on a general knowledge graph ignore the semantic relationship between each hop. However, modeling the knowledge base as a directed hypergraph has the problems of sparse incidence matrices and asymmetric Laplacian matrices. To make up for the deficiency, we propose a directed hypergraph convolutional network modeled on hyperbolic space, which can better deal with the sparse structure, and effectively adapt to the problem of an asymmetric incidence matrix of directed hypergraphs modeled on a knowledge base. We propose an interpretable KBQA model based on the hyperbolic directed hypergraph convolutional neural network named HDH-GCN which can update relation semantic information hop-by-hop and pays attention to different relations at different hops. The model can improve the accuracy of the multi-hop knowledge base question-answering task, and has application value in text question answering, human–computer interactions and other fields. Extensive experiments on benchmarks—PQL, MetaQA—demonstrate the effectiveness and universality of our HDH-GCN model, leading to state-of-the-art performance. [ABSTRACT FROM AUTHOR]

In this paper, we determine the maximum signless Laplacian spectral radius of all graphs which do not contain small books as a subgraph and characterize all extremal graphs. In addition, we give an upper bound of the signless Laplacian spectral radius of all graphs which do not contain intersecting quadrangles as a subgraph. [ABSTRACT FROM AUTHOR]

For a connected graph G on n vertices, recall that the reciprocal distance signless Laplacian matrix of G is defined to be R Q (G) = R T (G) + R D (G) , where R D (G) is the reciprocal distance matrix, R T (G) = d i a g (R T 1 , R T 2 , ⋯ , R T n) and R T i is the reciprocal distance degree of vertex v i . In 2022, generalized reciprocal distance matrix, which is defined by R D α (G) = α R T (G) + (1 − α) R D (G) , α ∈ [ 0 , 1 ] , was introduced. In this paper, we give some bounds on the spectral radius of R D α (G) and characterize its extremal graph. In addition, we also give the generalized reciprocal distance spectral radius of line graph L (G) . [ABSTRACT FROM AUTHOR]

This paper studies noise-to-state stability in probability (NSSP) for random complex dynamical systems on networks (RCDSN). On the basis of Kirchhoff's matrix theorem in graph theory, an appropriate Lyapunov function which combines with every subsystem for RCDSN is established. Moreover, some sufficient criteria closely related to the topological structure of RCDSN are given to guarantee RCDSN to meet NSSP by means of the Lyapunov method and stochastic analysis techniques. Finally, to show the usefulness and feasibility of theoretical findings, we apply them to random coupled oscillators on networks (RCON), and some numerical tests are given. [ABSTRACT FROM AUTHOR]

Let G be a connected, undirected and simple graph. The distance Laplacian matrix L (G) is defined as L (G) = d i a g (T r) − D (G) , where D (G) denotes the distance matrix of G and d i a g (T r) denotes a diagonal matrix of the vertex transmissions. Denote by ρ L (G) the distance Laplacian spectral radius of G. In this paper, we determine a lower bound of the distance Laplacian spectral radius of the n-vertex bipartite graphs with diameter 4. We characterize the extremal graphs attaining this lower bound. [ABSTRACT FROM AUTHOR]

S L E E has various applications in a large variety of problems. The signless Laplacian Estrada index of a hypergraph H is defined as S L E E (H) = ∑ i = 1 n e λ i (Q) , where λ 1 (Q) , λ 2 (Q) , ... , λ n (Q) are the eigenvalues of the signless Laplacian matrix of H. In this paper, we characterize the unique r-uniform unicyclic hypergraphs with maximum and minimum S L E E . [ABSTRACT FROM AUTHOR]

Spectral techniques are often used to partition the set of vertices of a graph, or to form clusters. They are based on the Laplacian matrix. These techniques allow easily to integrate weights on the edges. In this work, we introduce a p-Laplacian, or a generalized Laplacian matrix with potential, which also allows us to take into account weights on the vertices. These vertex weights are independent of the edge weights. In this way, we can cluster with the importance of vertices, assigning more weight to some vertices than to others, not considering only the number of vertices. We also provide some bounds, similar to those of Chegeer, for the value of the minimal cut cost with weights at the vertices, as a function of the first non-zero eigenvalue of the p-Laplacian (an analog of the Fiedler eigenvalue). [ABSTRACT FROM AUTHOR]

Unravelling how the human brain structure gives rise to function is a central question in neuroscience and remains partially answered. Recent studies show that the graph Laplacian of the human brain's structural connectivity (SC) plays a dominant role in shaping the pattern of resting-state functional connectivity (FC). The modeling of FC using the graph Laplacian of the brain's SC is limited, owing to the sparseness of the Laplacian matrix. It is unable to model the negative functional correlations. We extended the graph Laplacian to the hypergraph p-Laplacian in order to describe better the nonlinear and high-order relations between SC and FC. First we estimated those possible links showing negative correlations between the brain areas shared across subjects by statistical analysis. Then we presented a hypergraph p-Laplacian model by embedding the two matrices referring to the sign of the correlations between the brain areas relying on the brain structural connectome. We tested the model on two experimental connectome datasets and evaluated the predicted FC by estimating its Pearson correlation with the empirical FC matrices. The results showed that the proposed diffusion model based on hypergraph p-Laplacian can predict functional correlations more accurately than the models using graph Laplacian as well as hypergraph Laplacian. [ABSTRACT FROM AUTHOR]

Let G be a graph on n vertices. The Estrada index of G is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. V. Nikiforov studied hybrids of A (G) and D (G) and defined the A α -matrix for every real α ∈ [ 0 , 1 ] as: A α (G) = α D (G) + (1 − α) A (G). In this paper, using a different demonstration technique, we present a way to compare the Estrada index of the A α -matrix with the Estrada index of the adjacency matrix of the graph G. Furthermore, lower bounds for the Estrada index are established. [ABSTRACT FROM AUTHOR]

RNA stem-loop structures play an important role in almost every step of the viral replication cycle. In this contribution, a mathematical analysis is performed on a large dataset of RNA secondary structure elements in the coding regions of viruses by using topological indices that capture the Laplacian eigenvalues of the associated RNA graph representations and thereby enable structural classification, supplemented by folding energy and mutational robustness. The application of such an analysis for viral RNA structural motifs is described, being able to extract structural categories such as stem-loop structures of different sizes according to the tree-graph representation of the RNA structure, in our attempt to find novel functional motifs. While the analysis is carried on a large dataset of viral RNA structures, it can be applied more generally to other data that involve RNA secondary structures in biological agents. [ABSTRACT FROM AUTHOR]

Given a commutative ring R with identity 1 ≠ 0 , let the set Z (R) denote the set of zero-divisors and let Z * (R) = Z (R) \ { 0 } be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ (R) , is a simple graph whose vertex set is Z * (R) and each pair of vertices in Z * (R) are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs Γ (Z n) for n = p N 1 q N 2 , where p < q are primes and N 1 , N 2 are positive integers. [ABSTRACT FROM AUTHOR]

Fuzzy graph models enjoy the ubiquity of being present in nature and man-made structures, such as the dynamic processes in physical, biological, and social systems. As a result of inconsistent and indeterminate information inherent in real-life problems that are often uncertain, for an expert, it is highly difficult to demonstrate those problems through a fuzzy graph. Resolving the uncertainty associated with the inconsistent and indeterminate information of any real-world problem can be done using a vague graph (VG), with which the fuzzy graphs may not generate satisfactory results. The limitations of past definitions in fuzzy graphs have led us to present new definitions in VGs. The objective of this paper is to present certain types of vague graphs (VGs), including strongly irregular (SI), strongly totally irregular (STI), neighborly edge irregular (NEI), and neighborly edge totally irregular vague graphs (NETIVGs), which are introduced for the first time here. Some remarkable properties associated with these new VGs were investigated, and necessary and sufficient conditions under which strongly irregular vague graphs (SIVGs) and highly irregular vague graphs (HIVGs) are equivalent were obtained. The relation among strongly, highly, and neighborly irregular vague graphs was established. A comparative study between NEI and NETIVGs was performed. Different examples are provided to evaluate the validity of the new definitions. A new definition of energy called the Laplacian energy (LE) is presented, and its calculation is shown with some examples. Likewise, we introduce the notions of the adjacency matrix (AM), degree matrix (DM), and Laplacian matrix (LM) of VGs. The lower and upper bounds for the Laplacian energy of a VG are derived. Furthermore, this study discusses the VG energy concept by providing a real-time example. Finally, an application of the proposed concepts is presented to find the most effective person in a hospital. [ABSTRACT FROM AUTHOR]

The concept of smart cities emerged in the 1990s. Since then, smart buildings have become a closely interconnected element of smart cities. This type of building implements Internet of Things technology and control algorithms to monitor and control their indoor environment. The aim of this paper is to develop a new stability criterion method for smart building Internet of Things (IoT) systems, subject to external disturbances. The new stability criterion is going to optimize the operation of control algorithms since this criterion does not depend on the transmission function of the control algorithm but on the data collected by the IoT system. We present a new matrix called "Laplacian IoT matrix", containing IoT network information associated with the graph of a smart building. The proposal is supported by the results of a numerical case study. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n. The Estrada index and Laplacian Estrada index of G are defined by E E (G) = ∑ i = 1 n e λ i (A (G)) and L E E (G) = ∑ i = 1 n e λ i (L (G)) , where { λ i (A (G)) } i = 1 n and { λ i (L (G)) } i = 1 n are the eigenvalues of its adjacency and Laplacian matrices, respectively. In this paper, we establish almost sure upper bounds and lower bounds for random interdependent graph model, which is fairly general encompassing Erdös-Rényi random graph, random multipartite graph, and even stochastic block model. Our results unravel the non-triviality of interdependent edges between different constituting subgraphs in spectral property of interdependent graphs. [ABSTRACT FROM AUTHOR]

Let A (G) be the adjacent matrix and D (G) the diagonal matrix of the degrees of a graph G, respectively. For 0 ≤ α ≤ 1 , the A α -matrix is the general adjacency and signless Laplacian spectral matrix having the form of A α (G) = α D (G) + (1 − α) A (G) . Clearly, A 0 (G) is the adjacent matrix and 2 A 1 2 is the signless Laplacian matrix. A cactus is a connected graph such that any two of its cycles have at most one common vertex, that is an extension of the tree. The A α -spectral radius of a cactus graph with n vertices and k cycles is explored. The outcomes obtained in this paper can imply some previous bounds from trees to cacti. In addition, the corresponding extremal graphs are determined. Furthermore, we proposed all eigenvalues of such extremal cacti. Our results extended and enriched previous known results. [ABSTRACT FROM AUTHOR]

In this article, we find sharp lower bounds for the spectral radius of the distance signless Laplacian matrix of a simple undirected connected graph and we apply these results to obtain sharp upper bounds for the distance signless Laplacian energy graph. The graphs for which those bounds are attained are characterized. [ABSTRACT FROM AUTHOR]

Focusing first on the case HT ht , they determine bounds on the number of edges. Finally, the case of a fixed value HT ht is also discussed. They determine the graphs with largest HT ht -spectral radius with fixed vertex or edge connectivity. In addition, the corresponding extremal graphs are given and equations satisfying the HT ht -spectral radius are derived. [Extracted from the article]

Given a simple connected graph G, let D (G) be the distance matrix, D L (G) be the distance Laplacian matrix, D Q (G) be the distance signless Laplacian matrix, and T r (G) be the vertex transmission diagonal matrix of G. We introduce the generalized distance matrix D α (G) = α T r (G) + (1 − α) D (G) , where α ∈ [ 0 , 1 ] . Noting that D 0 (G) = D (G) , 2 D 1 2 (G) = D Q (G) , D 1 (G) = T r (G) and D α (G) − D β (G) = (α − β) D L (G) , we reveal that a generalized distance matrix ideally bridges the spectral theories of the three constituent matrices. In this paper, we obtain some sharp upper and lower bounds for the generalized distance energy of a graph G involving different graph invariants. As an application of our results, we will be able to improve some of the recently given bounds in the literature for distance energy and distance signless Laplacian energy of graphs. The extremal graphs of the corresponding bounds are also characterized. [ABSTRACT FROM AUTHOR]

The generalized distance matrix D α (G) of a connected graph G is defined as D α (G) = α T r (G) + (1 − α) D (G) , where 0 ≤ α ≤ 1 , D (G) is the distance matrix and T r (G) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α (G) . Some new upper and lower bounds for the generalized distance energy E D α (G) of G are established based on parameters including the Wiener index W (G) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n. [ABSTRACT FROM AUTHOR]

In this work, a general class of discrete time bidirectional associative memory (BAM) neural networks (NNs) is investigated. In this model, discrete and continuously distributed time delays are taken into account. By utilizing this novel method, which incorporates the approach of Kirchhoff's matrix tree theorem in graph theory, Continuation theorem in coincidence degree theory and Lyapunov function, we derive a few sufficient conditions to ensure the existence, uniqueness and exponential stability of the periodic solution of the considered model. At the end of this work, we give a numerical simulation that shows the effectiveness of this work. [ABSTRACT FROM AUTHOR]

In high-dimensional gene expression data analysis, the accuracy and reliability of cancer classification and selection of important genes play a very crucial role. To identify these important genes and predict future outcomes (tumor vs. non-tumor), various methods have been proposed in the literature. But only few of them take into account correlation patterns and grouping effects among the genes. In this article, we propose a rank-based modification of the popular penalized logistic regression procedure based on a combination of ℓ 1 and ℓ 2 penalties capable of handling possible correlation among genes in different groups. While the ℓ 1 penalty maintains sparsity, the ℓ 2 penalty induces smoothness based on the information from the Laplacian matrix, which represents the correlation pattern among genes. We combined logistic regression with the BH-FDR (Benjamini and Hochberg false discovery rate) screening procedure and a newly developed rank-based selection method to come up with an optimal model retaining the important genes. Through simulation studies and real-world application to high-dimensional colon cancer gene expression data, we demonstrated that the proposed rank-based method outperforms such currently popular methods as lasso, adaptive lasso and elastic net when applied both to gene selection and classification. [ABSTRACT FROM AUTHOR]

Highlights from the article: Imran et al. [[5]] deal with the metric dimension of special graphs, which is defined as the minimum cardinality of a resolving set of the corresponding graph. Gao et al. [[10]] consider another graph-theoretic subject, namely they deal with the extremal graphs of some topological indices of a graph I G i with given vertex I k i -partiteness, which is defined as the smallest number of vertices, the deletion of which from I G i yields a I k i -partite graph. In particular, they determine among the set of unicyclic graphs (these are graphs containing exactly one cycle) those graphs having the second-largest resistance-Harary index and, among the fully-loaded unicyclic graphs (these are unicyclic graphs such that no vertex with a degree less than three is in the unique cycle), those with the largest resistance-Harary index. 11 Zhang L., Zhao J., Liu J.-B., Arockiaraj M. Resistence Distance in H-Join of Graphs G1, G2, ..., Gk. Mathematics.

Calculating nodal voltages and branch current flows in a meshed network is fundamental to electrical engineering. This work demonstrates how such calculations can be performed using the eigenvalues and eigenvectors of the Laplacian matrix which describes the connectivity of the electrical network. These insights should permit the functioning of electrical networks to be understood in the context of spectral analysis. [ABSTRACT FROM AUTHOR]

For a graph G, the resistance distance r G (x , y) is defined to be the effective resistance between vertices x and y, the multiplicative degree-Kirchhoff index R ∗ (G) = ∑ { x , y } ⊂ V (G) d G (x) d G (y) r G (x , y) , where d G (x) is the degree of vertex x, and V (G) denotes the vertex set of G. L. Feng et al. obtained the element in C a c t (n ; t) with first-minimum multiplicative degree-Kirchhoff index. In this paper, we first give some transformations on R ∗ (G) , and then, by these transformations, the second-minimum multiplicative degree-Kirchhoff index and the corresponding extremal graph are determined, respectively. [ABSTRACT FROM AUTHOR]

Let (M , F , d μ) be a Finsler manifold with the Ricci curvature bounded below by a positive number and constant S-curvature. We prove that, if the first eigenvalue of the Finsler–Laplacian attains its lower bound, then M is isometric to a Finsler sphere. Moreover, we establish a comparison result on the Hessian trace of the distance function. [ABSTRACT FROM AUTHOR]

In view of the wide application of resistance distance, the computation of resistance distance in various graphs becomes one of the main topics. In this paper, we aim to compute resistance distance in H-join of graphs G 1 , G 2 , ... , G k . Recall that H is an arbitrary graph with V (H) = { 1 , 2 , ... , k } , and G 1 , G 2 , ... , G k are disjoint graphs. Then, the H-join of graphs G 1 , G 2 , ... , G k , denoted by ⋁ H { G 1 , G 2 , ... , G k } , is a graph formed by taking G 1 , G 2 , ... , G k and joining every vertex of G i to every vertex of G j whenever i is adjacent to j in H. Here, we first give the Laplacian matrix of ⋁ H { G 1 , G 2 , ... , G k } , and then give a { 1 } -inverse L (⋁ H { G 1 , G 2 , ... , G k }) { 1 } or group inverse L (⋁ H { G 1 , G 2 , ... , G k }) # of L (⋁ H { G 1 , G 2 , ... , G k }) . It is well know that, there exists a relationship between resistance distance and entries of { 1 } -inverse or group inverse. Therefore, we can easily obtain resistance distance in ⋁ H { G 1 , G 2 , ... , G k } . In addition, some applications are presented in this paper. [ABSTRACT FROM AUTHOR]

The vertex k-partiteness of graph G is defined as the fewest number of vertices whose deletion from G yields a k-partite graph. In this paper, we characterize the extremal value of the reformulated first Zagreb index, the multiplicative-sum Zagreb index, the general Laplacian-energy-like invariant, the general zeroth-order Randić index, and the modified-Wiener index among graphs of order n with vertex k-partiteness not more than m . [ABSTRACT FROM AUTHOR]