Search

Let D(G) and A(G), respectively, be the diagonal matrix of vertex de- grees and the adjacency matrix of a connected graph G. The generalized adjacency matrix of G is defined as Aα(G) = αD(G) + (1 − α)A(G), α ∈ [0, 1]. The spread of the generalized adjacency matrix, denoted by S(Aα(G)), is defined as the difference of the largest and smallest eigenvalues of Aα(G). The spread S(Q(G)) of Q(G), the signless Laplacian matrix of G and S(A(G)) of A(G) are similarly defined. In this paper, we establish the relationships between S(Aα(G)), S(Q(G)) and S(A(G)). Further, we find bounds for S(Aα(G)) involving different parameters of G. Also, we obtain lower bounds for S(Aα(G)) in terms of the clique number and independence number of G, and we characterize the extremal graphs in some cases. [ABSTRACT FROM AUTHOR]

The research on node importance identification for temporal networks has attracted much attention. In this work, combined with the multi-layer coupled network analysis method, an optimized supra-adjacency matrix (OSAM) modeling method was proposed. In the process of constructing an optimized super adjacency matrix, the intra-layer relationship matrixes were improved by introducing the edge weight. The inter-layer relationship matrixes were formed by improved similarly and the inter-layer relationship is directional by using the characteristics of directed graphs. The model established by the OSAM method accurately expresses the structure of the temporal network and considers the influence of intra- and inter-layer relationships on the importance of nodes. In addition, an index was calculated by the average of the sum of the eigenvector centrality indices for a node in each layer and the node importance sorted list was obtained from this index to express the global importance of nodes in temporal networks. The experimental results on three real temporal network datasets Enron, Emaildept3, and Workspace showed that compared with the SAM and the SSAM methods, the OSAM method has a faster message propagation rate and larger message coverage and better SIR and NDCG@10 indicators. [ABSTRACT FROM AUTHOR]

The extended adjacency matrix Ae (G) of a graph G is determined by graph degrees. Some inequalities are found for the extended adjacency matrix including its vertices, its edges and its degrees in this paper. Also, some bounds are established for this matrix involving its energy and its laplacian energy. [ABSTRACT FROM AUTHOR]

In a deregulated electric power system, it is important to distribute transmission losses between market participants fairly. This paper presents an effective method for active loss allocation by tracing power flow. In this method, after obtaining the alternating current (AC) load flow results of the system, it is first converted to a system in which the power flow is constant along each branch. A new matrix (i.e., power adjacency matrix) is then constructed for the system. The matrix is decoupled, and matrix operations are carried out using the submatrices to derive three other matrices. The contributions of the generators and loads of the main system to active losses of its branches and the shares of the generators in active power consumption of the loads are specified using these matrices. The proposed method is implemented on a four-bus system and several standard test systems. The results show that the method can be applied to any transmission system, and the loss allocation results are preferable to that of previous methods. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V(G) = {v1,v2,..., vn}. Let A(G) be the adjacency matrix of a graph G. The rows of A(G) corresponding to a vertex v of G, denoted by s(v) is the string. The Hamming index of a graph G is the sum of the Hamming distances between all pairs of vertices of G. In this paper we obtain Hamming index generated by adjacency matrix of some thorn graphs. [ABSTRACT FROM AUTHOR]

A mixed graph is a graph with edges and arcs, which can be considered as a hybrid of an undirected graph and a directed graph. In this paper we define the mixed adjacency matrix and the mixed energy of a mixed graph. The mixed adjacency matrix generalizes both the adjacency matrix of an undirected graph and the skew-adjacency matrix of a digraph. Then we compute the characteristic polynomial of the mixed adjacency matrix of a mixed graph and deduce some basic results from it. Furthermore, we give bounds to the mixed energy of a general mixed graph, and we compute the mixed energy of some special mixed graphs. At the end of the paper, we introduce mixed unitary Cayley graphs and compute their spectra. [ABSTRACT FROM AUTHOR]

Abstract: A threshold graph on n vertices is coded by a binary string of length . We obtain a formula for the inertia of (the adjacency matrix of) a threshold graph in terms of the code of the graph. It is shown that the number of negative eigenvalues of the adjacency matrix of a threshold graph is the number of ones in the code, whereas the nullity is given by the number of zeros in the code that are preceded by either a zero or a blank. A formula for the determinant of the adjacency matrix of a generalized threshold graph and the inverse, when it exists, of the adjacency matrix of a threshold graph are obtained. Results for antiregular graphs follow as special cases. [Copyright &y& Elsevier]

In this paper, we present an innovative transmit power control scheme, based on optimization theory, for wireless sensor networks (WSNs) which use carrier sense multiple access (CSMA) with collision avoidance (CA) as medium access control (MAC) protocol. In particular, we focus on schemes where several remote nodes send data directly to a common access point (AP). Under the assumption of finite overall network transmit power and low traffic load, we derive the optimal transmit power allocation strategy that minimizes the packet error rate (PER) at the AP. This approach is based on modeling the CSMA/CA MAC protocol through a finite state machine and takes into account the network adjacency matrix, depending on the transmit power distribution and determining the network connectivity. It will be then shown that the transmit power allocation problem reduces to a convex constrained minimization problem. Our results show that, under the assumption of low traffic load, the power allocation strategy, which guarantees minimal delay, requires the maximization of network connectivity, which can be equivalently interpreted as the maximization of the number of non-zero entries of the adjacency matrix. The obtained theoretical results are confirmed by simulations for unslotted Zigbee WSNs. [ABSTRACT FROM AUTHOR]

Abstract: Let be an unweighted rooted tree of k levels such that in each level the vertices have equal degree. Let d k−j+1 denotes the degree of the vertices in the level j. We find the eigenvalues of the adjacency matrix and of the Laplacian matrix of . They are the eigenvalues of principal submatrices of two nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for both matrices are , and , while the diagonal entries are zeros, in the case of the adjacency matrix, and d j , 1⩽ j ⩽k, in the case of the Laplacian matrix. Moreover, we give some results concerning to the multiplicity of the above mentioned eigenvalues. [Copyright &y& Elsevier]

Abstract: Let be the smallest number such that the adjacency matrix of any undirected graph with vertices or more has at least nonpositive eigenvalues. We show that is well-defined and prove that the values of for are 1, 3, 6, 10, 16 respectively. In addition, we prove that for all , , in which is the Ramsey number for and , and is the triangular number. This implies new lower bounds for eigenvalues of Laplacian matrices: the largest eigenvalue is bounded from below the largest degree, which generalizes some prior results. [Copyright &y& Elsevier]

Abstract: Let J be the all-ones matrix, and let A denote the adjacency matrix of a graph. An old result of Johnson and Newman states that if two graphs are cospectral with respect to yJ − A for two distinct values of y, then they are cospectral for all y. Here we will focus on graphs cospectral with respect to yJ − A for exactly one value of y. We call such graphs -cospectral. It follows that is a rational number, and we prove existence of a pair of -cospectral graphs for every rational . In addition, we generate by computer all -cospectral pairs on at most nine vertices. Recently, Chesnokov and the second author constructed pairs of -cospectral graphs for all rational , where one graph is regular and the other one is not. This phenomenon is only possible for the mentioned values of , and by computer we find all such pairs of -cospectral graphs on at most eleven vertices. [Copyright &y& Elsevier]

Let G be a graph. An edge orientation of G is called smooth if the in-degree and the out-degree of every vertex differ by at most one. In this paper, we show that if G is a 2-edge-connected non-bipartite graph with δ ( G ) ≥ 3 , then G has a smooth primitive orientation. Among other results, using the spectral radius of digraphs, we show that if D 1 is a primitive regular orientation and D 2 is a non-regular orientation of a given graph, then for sufficiently large t , the number of closed walks of length t in D 1 is more than the number of closed walks of length t in D 2 . [ABSTRACT FROM AUTHOR]

Abstract: Let G =(V, E) be a simple graph and {λ 1(G),…, λ n (G)} be its adjacency spectrum. It is easy to see that if an edge is added between two isolated vertices, then one zero eigenvalue increases by 1, and another zero eigenvalue decreases by 1. Let G + be a connected graph obtained from G by adding an edge e ∉ E(G). In this paper, it will be proved that the spectrum of G + is different from that of G only in two places with one eigenvalue increases by m and another eigenvalue decreases by m, where m >0 is a rational number, if and only if G is an empty graph with order 2. It will also be proved that one cannot construct a new adjacency integral connected graph with order n ⩾3 from a known one by adding an edge. [Copyright &y& Elsevier]

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]

Recently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we consider one type of directed graph. Then we obtain a general form of the adjacency matrices of the graph. By using the well-known property which states the (i, j) entry of Am (A is adjacency matrix) is equal to the number of walks of length m from vertex i to vertex j, we show that elements of mth positive integer power of the adjacency matrix correspond to well-known Jacobsthal numbers. As a consequence, we give a Cassini-like formula for Jacobsthal numbers. We also give a matrix whose permanents are Jacobsthal numbers. [ABSTRACT FROM AUTHOR]

The maximum eigenvalue of the adjacency matrix (AM) has been supposed to contain rich information about the corresponding network. An experimental study focused on revealing the meaning and application of the maximum eigenvalue is missing. To this end, AM was constructed using mutual information (MI) to determine the functional connectivity with electroencephalogram (EEG) data recorded with a mental fatigue model, and then was converted into both binary and weighted brain functional network (BFN) and corresponding random networks (RNs). Both maximum eigenvalue and corresponding network characters in BFNs and RNs were considered to explore the changes during the formation of mental fatigue. The results indicated that large maximum eigenvalue means more edges in the corresponding network, along with a high degree and a short characteristic path length both in weighted and binary BFNs. Interestingly, the maximum eigenvalue of AM was always a little larger than that of the corresponding random matrix (RM), and had an obvious linearity with the sum of the AM elements, indicating that the maximum eigenvalue can be able to distinguish the network structures which have the same mean degree. What is more, the maximum eigenvalue, which increased with the deepening of mental fatigue, can become a good indicator for mental fatigue estimation. [ABSTRACT FROM AUTHOR]

Abstract: We show that the adjacency matrix M of the line digraph of a d-regular digraph D on n vertices can be written as , where the matrix A is the Kronecker product of the all-ones matrix of dimension d with the identity matrix of dimension n and the matrix B is the direct sum of the adjacency matrices of the factors in a dicycle factorization of D. [Copyright &y& Elsevier]

Detecting communities in real world networks is an important problem for data analysis in science and engineering. By clustering nodes intelligently, a recursive algorithm is designed to detect community. Since the relabeling of nodes does not alter the topology of the network, the problem of community detection corresponds to the finding of a good labeling of nodes so that the adjacency matrix form blocks. By putting a fictitious interaction between nodes, the relabeling problem becomes one of energy minimization, where the total energy of the network is defined by putting interaction between the labels of nodes so that clustering nodes that are in the same community will decrease the total energy. A greedy method is used for the computation of minimum energy. The method shows efficient detection of community in artificial as well as real world network. The result is illustrated in a tree showing hierarchical structure of communities on the basis of sub-matrix density. Applications of the method to weighted and directed networks are discussed. [ABSTRACT FROM AUTHOR]

Abstract: We define cut-and-paste, a construction which, given a quadriculated disk obtains a disjoint union of quadriculated disks of smaller total area. We provide two examples of the use of this procedure as a recursive step. Tilings of a disk receive a parity: we construct a perfect or near-perfect matching of tilings of opposite parities. Let be the black-to-white adjacency matrix: we factor , where and are lower and upper triangular matrices, is obtained from a larger identity matrix by removing rows and columns and all entries of , and are equal to , or . [Copyright &y& Elsevier]

Abstract: Oscillations are a common type of plant-wide disturbances whose effects propagate to many units and thus may impact overall process performance. It is important to detect and diagnose such oscillations early in order to rectify the situation. Many frequency domain tools such as the power spectrum and spectrum envelope methods are capable of detecting the oscillation frequency. However, few methods are available for locating the root cause which is the main objective of oscillation diagnosis. This paper proposes a new method to diagnose the root cause of plant-wide oscillations using the adjacency matrix. A novel feature of the new method is that it utilizes the information in the process flowsheet. The method is not data-based and it can be carried out without using any data. However, this method complements the data based methods very well and it is best used in combination with other data-based methods to provide powerful diagnosis of plant-wide oscillations. This paper is the subject of a newly proposed complete procedure for detection and diagnosis of plant-wide oscillation. Two industrial case studies are also presented to demonstrate the applicability of the proposed procedure. [Copyright &y& Elsevier]

Let J(λ;n1,…,nk) be the set of matrices A such that λ is an eigenvalue of A and n1⩽⋯⩽nk are the sizes of the Jordan blocks associated with λ. For a given index v of A, denote by A−v the principal submatrix of co-order one obtained from A by deleting the vth row and column. In the present paper, all possible changes of the part of the Jordan form corresponding to λ under the transition from A to A−v are determined for matrices A∈J(λ;n1,…,nk) such that for the eigenvalue λ of both A and A⊤, there exists a Jordan chain of the largest length nk whose eigenvector has nonzero vth entry. In particular, it is shown that for almost every matrix A∈J(λ;n1,…,nk), n1,…,nk−1 are the sizes of Jordan blocks for λ considered as an eigenvalue of A−v. Moreover, it is also proved that if A is the adjacency matrix of a vertex-transitive digraph and k⩾2, then the change n1,…,nk→n1,…,nk−2,2nk−1−1 holds for the eigenvalue λ under the transition from A to A−v. In the case of k=1, λ is a simple eigenvalue of A and does not belong to the spectrum of A−v. [Copyright &y& Elsevier]

For a simple finite graph G, the generalized adjacency matrix is defined as A α (G) = α D (G) + (1 − α) A (G) , α ∈ [ 0 , 1 ] , where A (G) and D (G) are respectively the adjacency matrix and diagonal matrix of the vertex degrees. The A α -spread of a graph G is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the A α (G) . In this paper, we answer the question posed in (Lin, Z.; Miao, L.; Guo, S. Bounds on the A α -spread of a graph. Electron. J. Linear Algebra2020, 36, 214–227). Furthermore, we show that the path graph, P n , has the smallest S (A α) among all trees of order n. We establish a relationship between S (A α) and S (A). We obtain several bounds for S (A α) . [ABSTRACT FROM AUTHOR]

Biclustering algorithm is an effective tool for processing gene expression datasets. There are two kinds of data matrices, binary data and non-binary data, which are processed by biclustering method. A binary matrix is usually converted from pre-processed gene expression data, which can effectively reduce the interference from noise and abnormal data, and is then processed using a biclustering algorithm. However, biclustering algorithms of dealing with binary data have a poor balance between running time and performance. In this paper, we propose a new biclustering algorithm called the Adjacency Difference Matrix Binary Biclustering algorithm (AMBB) for dealing with binary data to address the drawback. The AMBB algorithm constructs the adjacency matrix based on the adjacency difference values, and the submatrix obtained by continuously updating the adjacency difference matrix is called a bicluster. The adjacency matrix allows for clustering of gene that undergo similar reactions under different conditions into clusters, which is important for subsequent genes analysis. Meanwhile, experiments on synthetic and real datasets visually demonstrate that the AMBB algorithm has high practicability. [ABSTRACT FROM AUTHOR]

A mixed graph D is a graph that can be obtained from a graph by orienting some of its edges. Let α be a primitive nth root of unity, then the α−Hermitian adjacency matrix of a mixed graph is defined to be the matrix Hα = [hrs] where hrs = α if rs is an arc in D, hrs = α if sr is an arc in D, hrs = 1 if sr is a digon in D and hrs = 0 otherwise. In this paper we study the cospectrality of the Hermitian adjacency matrix of a mixed graph. [ABSTRACT FROM AUTHOR]

Let D be a digraph of order n and let A(D) be the adjacency matrix of D. Let Deg (D) be the diagonal matrix of vertex out-degrees of D. For any real α ∈ [0, 1], the generalized adjacency matrix Aα(D) of the D is defined as Aα(D) = α Deg (D) + (1 - α)A(D). The largest modulus of the eigenvalues of Aα(D) is called the generalized adjacency spectral radius or the Aα-spectral radius of D. In this paper, we obtain some new upper and lower bounds for the spectral radius of Aα(D) in terms of the number of vertices n, the number of arcs, the vertex out-degrees, the average 2-out-degrees of the vertices of D and the parameter α. We characterize the extremal digraphs attaining these bounds. [ABSTRACT FROM AUTHOR]

The traditional modelling and simulation analysis methods based on graph theory are difficult to present the structural characteristics and dynamic laws of cyber-physical power systems (CPPS). In order to reduce the complexity of the attack graph, this paper proposes a CPPS robustness analysis method based on the reachable matrix. Based on the evolution of the augmented adjacency matrix of cyber-physical power system, this method uses the reachable matrix to obtain the maximum connectivity matrix of the system (the adjacency matrix corresponding to the largest connected subgraph of cyber-physical power system), and then, evaluates and analyses the robustness of cyber-physical power system. The IEEE118 node standard model is taken as an example to carry out the simulation experiment. The experimental results show that the reachable matrix has feasibility in the robustness analysis of cyber-physical power systems. Further through the contrast experiment found that compared with the robustness analysis method based on graph theory, the proposed method in this paper can reduce the modelling complexity of the mutual coupling network, reduce the complexity of the attack graph, and clearly and intuitively show the structural characteristics and dynamic laws of the network connection in the system. [ABSTRACT FROM AUTHOR]

Community structure is a significant characteristic of complex networks, and community detection has valuable applications in network structure analysis. Non-negative matrix factorization (NMF) is a key set of algorithms used to solve the community detection issue. Nevertheless, the localization of feature vectors in the adjacency matrix, which represents the characteristics of complex network structures, frequently leads to the failure of NMF-based approaches when the data matrix has a low density. This paper presents a novel algorithm for detecting sparse network communities using non-negative matrix factorization (NMF). The algorithm utilizes local feature vectors to represent the original network topological features and learns regularization matrices. The resulting feature matrices effectively reveal the global structure of the data matrix, demonstrating enhanced feature expression capabilities. The regularized data matrix resolves the issue of localized feature vectors caused by sparsity or noise, in contrast to the adjacency matrix. The approach has superior accuracy in detecting community structures compared to standard NMF-based community detection algorithms, as evidenced by experimental findings on both simulated and real-world networks. [ABSTRACT FROM AUTHOR]

In an earlier paper, we have introduced the Tosha-degree of an edge in a graph without multiple edges and studied some properties. In this paper, we extend the definition of Tosha-degree of an edge in a graph in which multiple edges are allowed. Also, we introduce the concepts - zero edges in a graph, T-line graph of a multigraph, Tosha-adjacency matrix, Tosha-energy, edge- adjacency matrix and edge energy of a graph G and obtain some results. [ABSTRACT FROM AUTHOR]

For a graph G the degree sum adjacency matrix DSA(G) is defined as a matrix, in which every element is sum of the degrees of the vertices if and only if the corresponding vertices are adjacent, otherwise it is zero. In this paper we obtain the bounds for the spectral radius and partial sum of the eigenvalues of the DSA matrix. We also find the bounds for the DSA energy of a graph in terms of its Zagreb indices. [ABSTRACT FROM AUTHOR]

Graphs can be associated with a matrix according to some rule and we can find the spectrum of a graph with respect to that matrix. Two graphs are cospectral if they have the same spectrum. Constructions of cospectral graphs help us establish patterns about structural information not preserved by the spectrum. We generalize a construction for cospectral graphs previously given for the distance Laplacian matrix to a larger family of graphs. In addition, we show that with appropriate assumptions this generalized construction extends to the adjacency matrix, combinatorial Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, and distance matrix. We conclude by enumerating the prevelance of this construction in small graphs for the adjacency matrix, combinatorial Laplacian matrix, and distance Laplacian matrix. [ABSTRACT FROM AUTHOR]

In today's world, various approaches and parameters exist for designing a plan and determining its spatial, placement. Hence, various modes for identifying crucial locations can be explored when an architectural plan is designed in different dimensions. While designing all these modes takes considerable time, there are numerous potential applications for artificial intelligence (AI) in this domain. This study aims to compute and use an adjacency matrix to generate architectural residential plans. Additionally, it develops a plan generation algorithm in Rhinoceros software, utilizing the Grasshopper plugin to create a dataset of architectural plans. In the following step, the data was entered into a neural network to identify the architectural plan's type, furniture, icons, and use of spaces, which was achieved using YOLOv4, EfficientDet, YOLOv5, DetectoRS, and RetinaNet. The algorithm's execution, testing, and training were conducted using Darknet and PyTorch. The research dataset comprises 12,000 plans, with 70% employed in the training phase and 30% in the testing phase. The network was appropriately trained practically and precisely in relation to an average precision (AP) resulting of 91.50%. After detecting the types of space use, the main research algorithm has been designed and coded, which includes determining the adjacency matrix of architectural plan spaces in seven stages. All research processes were conducted in Python, including dataset preparation, network object detection, and adjacency matrix algorithm design. Finally, the adjacency matrix is given to the input of the proposed plan generator network, which consequently, based on the resulting adjacency, obtains different placement modes for spaces and furniture. [ABSTRACT FROM AUTHOR]

For 0 ≤ α ≤ 1, Nikiforov proposed to study the spectral properties of the family of matrices Aα(G) = αD(G)+(1 - α)A(G) of a graph G, where D(G) is the degree diagonal matrix and A(G) is the adjacency matrix of G. The α-spectral radius of G is the largest eigenvalue of Aα(G). For a graph with two pendant paths at a vertex or at two adjacent vertices, we prove results concerning the behavior of the α-spectral radius under relocation of a pendant edge in a pendant path. We give upper bounds for the α-spectral radius for unicyclic graphs G with maximum degree ∆ ≥ 2, connected irregular graphs with given maximum degree and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with the second largest α-spectral radius among trees, and the unique tree with the largest α-spectral radius among trees with given diameter. We also determine the unique graphs so that the difference between the maximum degree and the α-spectral radius is maximum among trees, unicyclic graphs and non-bipartite graphs, respectively. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with the vertex set V = { v 1 , ... , v n } and denote by d v i the degree of the vertex v i . The modified Sombor index of G is the addition of the numbers (d v i 2 + d v j 2) − 1 / 2 over all of the edges v i v j of G. The modified Sombor matrix A M S (G) of G is the n by n matrix such that its (i , j) -entry is equal to (d v i 2 + d v j 2) − 1 / 2 when v i and v j are adjacent and 0 otherwise. The modified Sombor spectral radius of G is the largest number among all of the eigenvalues of A M S (G) . The sum of the absolute eigenvalues of A M S (G) is known as the modified Sombor energy of G. Two graphs with the same modified Sombor energy are referred to as modified Sombor equienergetic graphs. In this article, several bounds for the modified Sombor index, the modified Sombor spectral radius, and the modified Sombor energy are found, and the corresponding extremal graphs are characterized. By using computer programs (Mathematica and AutographiX), it is found that there exists only one pair of the modified Sombor equienergetic chemical graphs of an order of at most seven. It is proven that the modified Sombor energy of every regular, complete multipartite graph is 2 ; this result gives a large class of the modified Sombor equienergetic graphs. The (linear, logarithmic, and quadratic) regression analyses of the modified Sombor index and the modified Sombor energy together with their classical versions are also performed for the boiling points of the chemical graphs of an order of at most seven. [ABSTRACT FROM AUTHOR]

In an earlier work, the author together with Guo [7] introduced the Hermitian adjacency matrix of directed (and partially directed) graphs. However, it appears that a more natural Hermitian matrix exists, and it is the purpose of this note to bring this new Hermitian matrix to the attention of researchers in algebraic graph theory. [ABSTRACT FROM AUTHOR]

The Gershgorin Circle Theorem is a well-known and efficient method for bounding the eigenvalues of a matrix in terms of its entries. If A is a symmetric matrix, by writing A = B + x 1 , where 1 is the matrix with unit entries, we consider the problem of choosing x to give the optimal Gershgorin bound on the eigenvalues of B , which then leads to one-sided bounds on the eigenvalues of A. We show that this x can be found by an efficient linear program (whose solution can in may cases be written in closed form), and we show that for large classes of matrices, this shifting method beats all existing piecewise linear or quadratic bounds on the eigenvalues. We also apply this shifting paradigm to some nonlinear estimators and show that under certain symmetries this also gives rise to a tractable linear program. As an application, we give a novel bound on the lowest eigenvalue of a adjacency matrix in terms of the "top two" or "bottom two" degrees of the corresponding graph, and study the efficacy of this method in obtaining sharp eigenvalue estimates for certain classes of matrices. [ABSTRACT FROM AUTHOR]

The A γ matrix of a graph G is determined by A γ (G) = (1 − γ) A (G) + γ D (G) , where 0 ≤ γ ≤ 1 , A (G) and D (G) are the adjacency and the diagonal matrices of node degrees, respectively. In this case, the A γ matrix brings together the spectral theories of the adjacency, the Laplacian, and the signless Laplacian matrices, and many more γ adjacency-type matrices. In this paper, we obtain the A γ eigenvalues of zero divisor graphs of the integer modulo rings and the von Neumann rings. These results generalize the earlier published spectral theories of the adjacency, the Laplacian and the signless Laplacian matrices of zero divisor graphs. [ABSTRACT FROM AUTHOR]

To track the gradual change of the adjacency matrix of a simple graph G into the signless Laplacian matrix, V. Nikiforov in [35] suggested the study of the convex linear combination A α ( α-adjacency matrix ), A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) , for α ∈ [ 0 , 1 ] , where A ( G ) and D ( G ) are the adjacency and the diagonal vertex degrees matrices of G , respectively. Taking this definition as an idea the next matrix was considered for a , b ∈ R . The matrix A a , b defined by A a , b ( G ) = a D ( G ) + b A ( G ) , extends the previous α -adjacency matrix. This matrix is designated the ( a , b ) -adjacency matrix of G . Both adjacency matrices are examples of universal matrices already studied by W. Haemers. In this paper, we study the ( a , b ) -adjacency spectra for a family of compound graphs formed by disjoint balanced trees whose roots are identified to the vertices of a given graph. In consequence, new families of cospectral (adjacency, Laplacian and signless Laplacian) graphs, new hypoenergetic graphs (graphs whose energy is less than its vertex number) and new explicit formulae for Estrada, signless Laplacian Estrada and Laplacian Estrada indices of graphs were obtained. Moreover, sharp upper bounds of the above indices for caterpillars, in terms of length of the path and of the maximum number of its pendant vertices, are given. [ABSTRACT FROM AUTHOR]

For a vertex v in a graph G, the open support of v under addition is the sum of degrees of all its neighbors. The open support of G under addition is the sum of open supports of all its vertices. The results for open support of graphs are deeply dependent on the structure property of the graph considered, such as its symmetry. In this paper, we generalize the concept of open support to hypergraphs. We give some examples and prove a general formula for open support of hypergraphs by using the adjacency and incidence matrices method. [ABSTRACT FROM AUTHOR]

Abstract The spectral radius of a graph is the largest eigenvalue of the adjacency matrix corresponding to this graph. Let G be a graph with long pendant paths. In this note, we estimate the spectral radius of G , by obtaining appropriate upper and lower bounds. As an application, the spectral radius of some k-ary trees are estimated. [ABSTRACT FROM AUTHOR]

Abstract It is proved that for any finite connected graph G , there exists an orientation of G such that the spectral radius of the corresponding Hermitian adjacency matrix is smaller or equal to the spectral radius of the universal cover of G (with equality if and only if G is a tree). This provides a suitable answer to a problem proposed by Mohar. The proof uses the method of interlacing families of polynomials that was developed by Marcus, Spielman, and Srivastava in their seminal work on the existence of infinite families of Ramanujan graphs. [ABSTRACT FROM AUTHOR]

The numerical range of a simple graph G, named F(G), is the numerical range of its adjacency matrix A(G). The main purpose of this paper is to approximate F(G). Then, using this approximation, bounds for the largest and the smallest eigenvalues of G are proposed. In fact, lower bounds for the largest eigenvalues of G are presented in terms of disjoint induced subgraphs of G and the numerical range of the square of A(G). [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: The signless Laplacian Q(G) of G is defined as Q(G) := A(G) +D(G). Cvetković called the study of the adjacency matrix the A-spectral theory, and the study of the signless Laplacian-the Q-spectral theory. To track the gradual change of A(G) into Q(G), in this paper it is suggested to study the convex linear combinations Aα(G) of A(G) and D(G) defined byAα (G) := αD(G) + (1 - α )A(G) , 0 ≤ α ≤ 1. This study sheds new light on A(G) and Q(G), and yields, in particular, a novel spectral Turán theorem. A number of open problems are discussed. [ABSTRACT FROM AUTHOR]

In this note, I derive simple formulas based on the adjacency matrix of a network to compute measures associated with Ronald S. Burt's structural holes (effective size, redundancy, local constraint, and constraint), together with the measure called improved structural holes introduced in 2017. This can help to see these measures within a unified computation framework because they can all be expressed in matricial form. These formulas can also be used to define naïve algorithms based on matrix operations for their computation. Such naïve algorithms can be used for small- and medium-sized networks, where exploiting the sparsity of the matrices and efficient triangle listing techniques are not necessary. [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 [3] defined the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . The largest eigenvalue of A α ( G ) is called the A α -spectral radius of G . In this paper, we give three edge graft transformations on A α -spectral radius. As applications, we determine the unique graph with maximum A α -spectral radius among all connected graphs with diameter d , and determine the unique graph with minimum A α -spectral radius among all connected graphs with given clique number. In addition, some bounds on the A α -spectral radius are obtained. [ABSTRACT FROM AUTHOR]

In this work, we introduce balance-binomial graphs, whose entries of adjacency matrix are balance-binomial coefficients. Then we obtain some characteristic properties of the graph. [ABSTRACT FROM AUTHOR]

Let A = [aij ] be an n × n real symmetric matrix with eigenvalues λ1,..., λn. The energy of A, denoted by E(A), is defined as |λ1| + · · · + |λn|. We prove that if A is non-zero and |λ1| ≥ · · · ≥ |λn|, then E(A) ≥ n|λ1||λn| + P 1≤i,j≤n a 2 ij |λ1| + |λn| (0.1). In particular, we show that Ψ(A)E(A) ≥ P 1≤i,j≤n a 2 ij, where Ψ(A) is the maximum value of the sequence Pn j=1 |a1j |, Pn j=1 |a2j |, . . . ,Pn j=1 |anj |. The energy of a simple graph G, denoted by E(G), is defined as the energy of its adjacency matrix. As an application of inequality (0.1) we show that if G is a t-regular graph (t ̸= 0) of order n with no eigenvalue in the interval (-1, 1), then E(G) ≥ 2tn t+1 and the equality holds if and only if every connected component of G is the complete graph Kt+1 or the crown graph K*t+1. [ABSTRACT FROM AUTHOR]

Abstract: For a simple graph G, let rank(G) and dnzr(G) denote respectively the rank and the number of distinct nonzero rows of the adjacency matrix A(G) of G. Equivalent conditions are given for the join of two vertex-disjoint graphs to satisfy . A new proof is provided for the known relation rank(G)=dnzr(G) for cographs G. Our approach relies on the concepts of neighborhood equivalence classes, reduced graph and reduced adjacency matrix, and also on a known result that relates the spectrum of the adjacency matrix of a graph with that of its reduced adjacency matrix as well as a new characterization of the nonsingularity of a real symmetric matrix in a special 2×2 block form. Our treatment provides ways to construct graphs G, other than cographs, that satisfy rank(G)=dnzr(G). As a side result we also show that every rational number is equal to the sum of the entries of the inverse of the adjacency matrix of a connected nonsingular graph. [Copyright &y& Elsevier]

Hop Count Matrix (HCM) contains rich connectivity information, which is very important for various Internet of Things (IoT) applications, especially for obtaining the locations of sensor nodes. However, some items of HCMs may be missing due to attacks by malicious nodes or unexpected termination of flooding operations. To solve this problem, two methods, called HCMR-AM and HCMR-DT, are proposed to recover the missing items. In HCMR-AM, the collected partial hop counts are employed to construct Adjacency Matrix (AM), and then the constructed AM is used to obtain the complete HCM. In HCMR-DT, the recovery of HCM is transformed into a classification problem, where multi-dimensional features are used for joint prediction to achieve more accurate recovery performance. Extensive experimental results demonstrate that compared to the original SVT and HCMR-NBC, our proposed algorithms have significant improvement in recovery performance and execution efficiency. In addition, the complete HCM is used for node localization, and experimental results show that the HCM recovered by the proposed methods can achieve the same localization performance as the HCM without missing value when the observation ratio of HCM is greater than 30%, which cannot be achieved by other recovery algorithms. • A hop count matrix recovery scheme based on adjacency matrix is proposed. • A hop count matrix recovery method using Decision Tree is developed. • The complete hop count matrices are suitable for range-free localization. [ABSTRACT FROM AUTHOR]

Abstract: Consider a graph on vertices with adjacency matrix and degree sequence . A universal adjacency matrix of is any matrix in Span with a nonzero coefficient for , where and and are the identity and all-ones matrix, respectively. Thus a universal adjacency matrix is a common generalization of the adjacency, the Laplacian, the signless Laplacian and the Seidel matrix. We investigate graphs for which some universal adjacency matrix has just two eigenvalues. The regular ones are strongly regular, complete or empty, but several other interesting classes occur. [Copyright &y& Elsevier]

Background: There are many computational approaches to predict the protein-protein interactions using support vector machines (SVMs) with high performance. In fact, performance of currently reported methods are significantly over-estimated and affected by the object repetitiveness in the datasets used. Objective: To study the effect of object repetitiveness of datasets on predicting results. Method: We present novel methods to construct different positive datasets with or without repeating proteins using graph maximum matching in the protein-protein interaction datasets and corresponding series of negative datasets with different proteins repetitiveness are constructed using graph adjacency matrix. The relationship between the SVM prediction results and the repeated proteins (repeat numbers and repeat rates) and the distributions of repeated proteins in the datasets are analyzed. Results: Protein repetitiveness of positive and negative datasets can affect the prediction result: high protein repetitiveness of positive or negative datasets yield high performance prediction result. Conclusion: This indicate that dealing with object repetitiveness of datasets is a key issue in protein-protein interactions prediction using SVMs since real world data contain certain degrees of repeat proteins. [ABSTRACT FROM AUTHOR]