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Abstract: The spread of a graph is defined to be the difference between the largest eigenvalue and the least eigenvalue of the adjacency matrix of the graph. Let denote the set of connected unicyclic graphs of order n and girth k, and let denote the set of connected unicyclic graphs of order n. In this paper, we determine the unique graph with minimum least eigenvalue (respectively, the unique graph with maximum spread) among all graphs in . We, finally, characterize the unique graph with minimum least eigenvalue (respectively, the unique graph with maximum spread) among all graphs in . [Copyright &y& Elsevier]

Abstract: A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let be a set of trees such that, for [(1)] is a generalized Bethe tree of levels, [(2)] the vertices of at the level j have degree for , and [(3)] the edges of joining the vertices at the level j with the vertices at the level have weight for . Let be the tree obtained from the union of the trees joined at their respective root vertices. We give a complete characterization of the eigenvalues of the Laplacian and adjacency matrices of . Moreover, we derive results concerning their multiplicities. In particular, we characterize the spectral radii, the algebraic conectivity and the second largest Laplacian eigenvalue. [Copyright &y& Elsevier]

Abstract: Let G be a weighted rooted graph of k levels such that, for [(1)] each vertex at level j is adjacent to one vertex at level and all edges joining a vertex at level j with a vertex at level have the same weight, where the weight is a positive real number; [(2)] if two vertices at level j are adjacent then they are adjacent to the same vertex at level and all edges joining two vertices at level j have the same weight; [(3)] two vertices at level j have the same degree; [(4)] there is not a vertex at level j adjacent to others two vertices at the same level; We give a complete characterization of the eigenvalues of the Laplacian matrix and adjacency matrix of G. They are the eigenvalues of leading principal submatrices of two nonnegative symmetric tridiagonal matrices of order and the roots of some polynomials related with the characteristic polynomial of the referred submatrices. By application of the above mentioned results, we derive an upper bound on the largest eigenvalue of a graph defined by a weighted tree and a weighted triangle attached, by one of its vertices, to a pendant vertex of the tree. [Copyright &y& Elsevier]

Abstract: We characterize the graphs which achieve the maximum value of the spectral radius of the adjacency matrix in the sets of all graphs with a given domination number and graphs with no isolated vertices and a given domination number. [Copyright &y& Elsevier]

Abstract: In this paper, we study the spectral properties of a family of trees characterized by two main features: they are spanning subgraphs of the hypercube, and their vertices bear a high degree of (connectedness) hierarchy. Such structures are here called binary hypertrees and they can be recursively defined as the so-called hierarchical product of several complete graphs on two vertices. [Copyright &y& Elsevier]

Abstract: A graph G is singular of nullity , if its adjacency matrix is singular, with the eigenvalue zero of multiplicity . A singular graph having a 0-eigenvector, , with no zero entries, is called a core graph. We place particular emphasis on nut graphs, namely the core graphs of nullity one. Through symmetry considerations of the automorphism group of the graph, we study relations among the entries of which lead to interesting implications in chemistry. The zero eigenvalue is rare in a fullerene graph. We show that there are possible nut fullerenes with relatively simple structures. [Copyright &y& Elsevier]

Abstract: Let G be an n-vertex () simple graph embeddable on a surface of Euler genus (the number of crosscaps plus twice the number of handles). Denote by the maximum degree of G. In this paper, we first present two upper bounds on the Laplacian spectral radius of G as follows: [(i)] [(ii)] if G is 4-connected and either the surface is the sphere or the embedding is 4-representative, then Some upper bounds on the Laplacian spectral radius of the outerplanar and Halin graphs are also given. [Copyright &y& Elsevier]

Abstract: Let be a unicyclic simple undirected graph with largest vertex degree Δ. Let be the unique cycle of . The graph is a forest of r rooted trees with root vertices , respectively. Letwhere is the distance from v to u. Let and be the spectral radius of the Laplacian matrix and adjacency matrix of , respectively. We prove thatwhenever andwhenever or whenever and . [Copyright &y& Elsevier]

Abstract: A Bethe tree is a rooted unweighted of k levels in which the root vertex has degree equal to d, the vertices at level j have degree equal to and the vertices at level k are the pendant vertices. In this paper, we first derive an explicit formula for the eigenvalues of the adjacency matrix of . Moreover, we give the corresponding multiplicities. Next, we derive an explicit formula for the simple nonzero eigenvalues, among them the largest eigenvalue, of the Laplacian matrix of . Finally, we obtain upper bounds on the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any tree . These upper bounds are given in terms of the largest vertex degree and the radius of , and they are attained if and only if is a Bethe tree. [Copyright &y& Elsevier]

Abstract: Various types of LU-factorizations for nonsingular matrices, where L is a lower triangular matrix and U is an upper triangular matrix, are defined and characterized. These types of LU-factorizations are extended to the general m × n case. The more general conditions are considered in the light of the structures of [C.R. Johnson, D.D. Olesky, P. Van den Driessche, Inherited matrix entries: LU factorizations, SIAM J. Matrix Anal. Appl. 10 (1989) 99–104]. Applications to graphs and adjacency matrices are investigated. Conditions for the product of a lower and an upper triangular matrix to be the zero matrix are also obtained. [Copyright &y& Elsevier]

Abstract: For acyclic and unicyclic graphs we determine a necessary and sufficient condition for a graph G to be singular. Further, it is shown that this characterization can be used to construct a basis for the null-space of G. [Copyright &y& Elsevier]

Abstract: We prove the existence of equiangular tight frames having elements drawn from either or whenever n is either for , or a power of a prime such that . We also find a simple explicit expression for the prime power case by establishing a connection to a -element equiangular tight frame based on quadratic residues. [Copyright &y& Elsevier]

Severe thunderstorms are a manifestation of deep convection. Conditional instability is known to be the mechanism by which thunderstorms are formed. The energy that drives conditional instability is convective available potential energy (CAPE), which is computed with radio sonde data at each pressure level. The purpose of the present paper is to identify the pattern or shape of CAPE required for the genesis of severe thunderstorms over Kolkata (22°32′N, 88°20′E) confined within the northeastern part (20°N to 24°N latitude, 85°E to 93°E longitude) of India. The method of chaotic graph theory is adopted for this purpose. Chaotic graphs of pressure levels and CAPE are formed for thunderstorm and non-thunderstorm days. Ranks of the adjacency matrices constituted with the union of chaotic graphs of pressure levels and CAPE are computed for thunderstorm and non-thunderstorm days. The results reveal that the rank of the adjacency matrix is maximum for non-thunderstorm days and a column with all zeros occurs very quickly on severe thunderstorms days. This indicates that CAPE loses connectivity with pressure levels very early on severe thunderstorm days, showing that for the genesis of severe thunderstorms over Kolkata short, and therefore broad, CAPE is preferred. [ABSTRACT FROM AUTHOR]

Abstract: A set of vertices is -regular if it induces a k-regular subgraph of G such that . Note that a connected graph with more than one edge has a perfect matching if and only if its line graph has a -regular set. In this paper, some spectral results on the adjacency matrix of graphs with -regular sets are presented. Relations between the combinatorial structure of a p-regular graph with a -regular set and the eigenspace corresponding to each eigenvalue are deduced. Finally, additional results on the effects of Seidel switching (with respect to a bipartition induced by S) of regular graphs are also introduced. [Copyright &y& Elsevier]

Abstract: Pattern recognition techniques can be very useful in forensic sciences to point out to relevant sets of events and potentially encourage an intelligence-led style of policing. In this study, these techniques have been applied to categorical data corresponding to cutting agents found in heroin seizures. An application of graph theoretic methods has been performed, in order to highlight the possible relationships between the location of seizures and co-occurrences of particular heroin cutting agents. An analysis of the co-occurrences to establish several main combinations has been done. Results illustrate the practical potential of mathematical models in forensic data analysis. [Copyright &y& Elsevier]

Abstract: Let G be a graph of order n and rank(G) denotes the rank of its adjacency matrix. Clearly, . In this paper we characterize all graphs G such that or n +2. Also for every integer n ⩾5 and any k, 0⩽ k ⩽ n, we construct a graph G of order n, such that . [Copyright &y& Elsevier]

Abstract: Let G be a simple graph. Let and denote the largest eigenvalue of the adjacency matrix and the Laplacian matrix of G, respectively. Let denote the largest vertex degree. If G has just one cycle, thenThe equality holds if and only if . AndThe equality holds if and only if , n is even. [Copyright &y& Elsevier]

Abstract: The star complement technique is a spectral tool recently developed for constructing some bigger graphs from their smaller parts, called star complements. Here we first identify among trees and complete graphs those graphs which can be star complements for 1 as the second largest eigenvalue. Using the graphs just obtained, we next search for their maximal extensions, either by theoretical means, or by computer aided search. [Copyright &y& Elsevier]

Abstract: We generalize the concept of a Bethe tree as follows: we say that an unweighted rooted tree is a generalized Bethe tree if in each level the vertices have equal degree. If is a generalized Bethe tree of k levels then we characterize completely the eigenvalues of the adjacency matrix and Laplacian matrix of a graph obtained from the union of r copies of and the cycle connecting the r vertex roots. Moreover, we give results on the multiplicity of the eigenvalues, on the spectral radii and on the algebraic conectivity. [Copyright &y& Elsevier]

Abstract: Let be a weighted rooted tree of k levels such that [(1)] the vertices in level j have a degree equal to d k−j+1 for j =1,2,…, k, and [(2)] the edges joining the vertices in level j with the vertices in level (j +1) have a weight equal to w k−j for j =1,2,…, k−1. We give a complete characterization of the eigenvalues of the Laplacian matrix and adjacency matrix of . They are the eigenvalues of leading principal submatrices of two nonnegative symmetric tridiagonal matrices of order k × k. Moreover, we give some results concerning their multiplicities. By application of the above mentioned results, we derive upper bounds on the largest eigenvalue of any weighted tree and the spectra of some weighted Bethe trees. [Copyright &y& Elsevier]

This paper presents a new MSDI (mold surface data integration) method for presenting ruled mold-surface data for CAPP applications. The method makes use of adjacency and feature-data matrices to establish a framework of feature-based data. The MSDI method enables the storage, retrieval and operations of geometries, topologies, and machining information of a given ruled mold surface. The matrices of adjacency and feature-data effectively represent the relations. Boolean algebra and group technology (GT) methods are introduced to generate the desired process plans. A sample mold made of ruled surfaces is made and used to show the effectiveness of the proposed data representation and the integrated mold surface processing method. A compact computer program is implemented to enhance the speed and accuracy of the MSDI method. The numerical results indicate that the mold surface decomposition and integration procedure presented in this paper are systematic and effective. The paper provides a novel modular mold surface modeling tool and procedure that can be further extended to accommodate more features appearing in complicated ruled mold surfaces. [ABSTRACT FROM AUTHOR]

Abstract: The higher Randić index R t (G) of a simple graph G is defined aswhere δ i denotes the degree of the vertex i and i 1 i 2⋯i t+1 runs over all paths of length t in G. In [J.A. Rodríguez, A spectral approach to the Randić index, Linear Algebra Appl. 400 (2005) 339–344], the lower and upper bound on R 1(G) was determined in terms of a kind of Laplacian spectra, and the lower and upper bound on R 2(G) were done in terms of kinds of adjacency and Laplacian spectra. In this paper we characterize the graphs which achieve the upper or lower bounds of R 1(G) and R 2(G), respectively. [Copyright &y& Elsevier]

Abstract: In this note we study some properties of the spectrum of a connected graph G with four different eigenvalues, and (spectrally maximum) diameter three. When G is regular, this is the case, for instance, when G is a distance-regular graph. [Copyright &y& Elsevier]

Abstract: The conditional diameter of a connected graph is defined as follows: given a property of a pair of subgraphs of , the so-called conditional diameter or -diameter measures the maximum distance among subgraphs satisfying . That is,In this paper we consider the conditional diameter in which requires that for all , for all , and for some integers and , where denotes the degree of a vertex x of , denotes the minimum degree and the maximum degree of . The conditional diameter obtained is called -diameter. We obtain upper bounds on the -diameter by using the k-alternating polynomials on the mesh of eigenvalues of an associated weighted graph. The method provides also bounds for other parameters such as vertex separators. [Copyright &y& Elsevier]

Abstract: Let be an unweighted tree of k levels such that in each level the vertices have equal degree. Let n k−j+1 and d k−j+1 be the number of vertices and the degree of them in the level j. We find the eigenvalues of the adjacency matrix and Laplacian matrix of for the case of two vertices in level 1 (n k =2), including results concerning to their multiplicity. They are the eigenvalues of leading principal submatrices of nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for these matrices are , 2⩽ j ⩽ k, while the diagonal entries are 0,…,0,±1, in the case of the adjacency matrix, and d 1, d 2,…, d k−1, d k ±1, in the case of the Laplacian matrix. Finally, we use these results to find improved upper bounds for the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any given tree. [Copyright &y& Elsevier]

Abstract: We apply two methods to the block diagonalization of the adjacency matrix of the Cayley graph defined on the affine group. The affine group will be defined over the finite ring . The method of irreducible representations will allow us to find nontrivial eigenvalue bounds for two different graphs. One of these bounds will result in a family of Ramanujan graphs. The method of covering graphs will be used to block diagonalize the affine graphs using a Galois group of graph automorphisms. In addition, we will demonstrate the method of covering graphs on a generalized version of the graphs of Lubotzky et al. [A. Lubotzky, R. Phillips, P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988) 261–277]. [Copyright &y& Elsevier]

Abstract: Let be an unweighted tree with vertex root v which is the union of two trees , such that V 1 ∩ V 2 ={v} and and have the property that the vertices in each of their levels have equal degree. We characterize completely the eigenvalues of the adjacency matrix and of the Laplacian matrix of . They are the eigenvalues of symmetric tridiagonal matrices whose entries are given in terms of the vertex degrees. Moreover, we give some results about the multiplicity of the eigenvalues. Applications to some particular trees are developed. [Copyright &y& Elsevier]

Abstract: The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. F. Zhang, H. Li [On acyclic conjugated molecules with minimal energies, Discrete Appl. Math. 92 (1999) 71–84] characterized the trees with a perfect matching having the minimal and the second minimal energies, which solved a conjecture proposed by I. Gutman [Acyclic conjugated molecules, trees and their energies, J. Math. Chem. 1 (1987) 123–143]. In this letter, for a given positive integer we characterize the tree with the minimal energy having diameter at least . As a corollary, we also characterize the tree with the minimal Hosoya index having diameter at least . [Copyright &y& Elsevier]

Abstract: Let be a tree with vertex set V. Let d v denotes the degree of v ∈ V. Let Δ =max{d v : v ∈ V}. Let u ∈ V such that d u = Δ. Let k = e u +1 where e u is the excentricity of u. For j =1,2,…, k −2, letWe prove thatandwhere and are the largest eigenvalue of the Laplacian matrix and adjacency matrix of T, respectively. These bounds give better results than those obtained in [D. Stevanović, Linear Algebra Appl. 360 (2003) 35–42] except if δ 1 = Δ. [Copyright &y& Elsevier]

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]

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]

Something between an expository note and an extended research problem, this article is an invitation to expand the existing literature on a family of graph invariants rooted in linear and multilinear algebra. There are a variety of ways to assign a real n×n matrix K(G) to each n-vertex graph G, so that G and H are isomorphic if and only if K(G) and K(H) are permutation similar. It follows that G and H are isomorphic only if K(G) and K(H) are similar, i.e., that similarity invariants of K(G) are graph theoretic invariants of G, an observation that helps to explain the enormous literature on spectral graph theory. The focus of this article is the permutation part, i.e., on matrix functions that are preserved under permutation similarity if not under all similarity. [Copyright &y& Elsevier]

Abstract: This is a survey of recent results showing that the usual concepts of (local or global) distance-regularity in a graph can be thought of as an extremal (numeric) property of the graph. This is because such structures appear when a certain spectral bound is attained, so yielding striking characterizations of distance-regularity, with the peculiarity of involving only numerical (instead of the usual matricial) identities. Other results providing bounds for specific parameters of the graph, such as its eigenvalue multiplicities, are also derived. [Copyright &y& Elsevier]

Let G be a simple undirected graph. For v∈V(G), the 2-degree of v is the sum of the degrees of the vertices adjacent to v. Denote by ρ(G) and μ(G) the spectral radius of the adjacency matrix and the Laplacian matrix of G, respectively. In this paper, we present two lower bounds of ρ(G) and μ(G) in terms of the degrees and the 2-degrees of vertices. [Copyright &y& Elsevier]

The eigenvalues of a graph are the eigenvalues of its adjacency matrix. This paper presents some upper and lower bounds on the greatest eigenvalue and a lower bound on the smallest eigenvalue. [Copyright &y& Elsevier]

It is proved that any set of finite pairwise nonisomorphic weakly cospectral connected pseudographs, as well as any set of pseudodigraphs satisfying various additional conditions, is finite. [ABSTRACT FROM AUTHOR]

We introduce the concept of weakly distance-regular digraph and study some of its basic properties. In particular, the (standard) distance-regular digraphs, introduced by Damerell, turn out to be those weakly distance-regular digraphs which have a normal adjacency matrix. As happens in the case of distance-regular graphs, the study is greatly facilitated by a family of orthogonal polynomials called the distance polynomials. For instance, these polynomials are used to derive the spectrum of a weakly distance-regular digraph. Some examples of these digraphs, such as the butterfly and the cycle prefix digraph which are interesting for their applications, are analyzed in the light of the developed theory. Also, some new constructions involving the line digraph and other techniques are presented. [Copyright &y& Elsevier]

The early research on regular graphs of the French mathematician Georges Brunel (1856–1900) is discussed. Brunel developed early graph terminology and started the French “applied” approach towards graph theory. [Copyright &y& Elsevier]

It was proved recently by one of the authors that, if H is a path Pt (t>2 with t≠7 or 8) or an odd cycle Ct (t>3), then there is a unique maximal graph having H as a star complement for −2. The methods employed were analytical in nature, making use of the Reconstruction Theorem for star complements. Here we offer an alternative approach, based on the forbidden subgraph technique. In addition, we resolve the exceptional situations arising when H=P7 or P8. [Copyright &y& Elsevier]

Let D be a digraph, and let Ln(D) and k(D) denote the nth iterated line digraph of D and the index of convergence of D, respectively. We prove that if D contains neither sources nor sinks, then: (1) k(Ln(D))=k(D)=0 if every connected component of D is a directed cycle. (2) k(Ln(D))=k(D)+n if there exists at least one connected component of D that is not a directed cycle. Finally, we prove that if D has no sources or no sinks then k(D)⩽k(L(D))⩽k(D)+1. [Copyright &y& Elsevier]

New classes of random graphs have recently been shown to exhibit the small world phenomenon—they are clustered like regular lattices and yet have small average pathlengths like traditional random graphs. Small world behaviour has been observed in a number of real life networks, and hence these random graphs represent a useful modelling tool. In particular, Grindrod [Phys. Rev. E 66 (2002) 066702-1] has proposed a class of range dependent random graphs for modelling proteome networks in bioinformatics. A property of these graphs is that, when suitably ordered, most edges in the graph are short-range, in the sense that they connect near-neighbours, and relatively few are long-range. Grindrod also looked at an inverse problem—given a graph that is known to be an instance of a range dependent random graph, but with vertices in arbitrary order, can we reorder the vertices so that the short-range/long-range connectivity structure is apparent? When the graph is viewed in terms of its adjacency matrix, this becomes a problem in sparse matrix theory: find a symmetric row/column reordering that places most nonzeros close to the diagonal. Algorithms of this general nature have been proposed for other purposes, most notably for reordering to reduce fill-in and for clustering large data sets. Here, we investigate their use in the small world reordering problem. Our numerical results suggest that a spectral reordering algorithm is extremely promising, and we give some theoretical justification for this observation via the maximum likelihood principle. [Copyright &y& Elsevier]

Let λ1(G) denote the largest eigenvalue of the adjacency matrix and let μ1(G) denote the largest eigenvalue of the Laplacian matrix of a graph G. It is well known that if a graph G has the largest vertex degree Δ≠0 then√ of Δ&les;λ1(G)&les;ΔandΔ+1&les;μ1(G)&les;2Δ.Thus the gap between the maximum and minimum value of λ1(G) and μ1(G) in the class of graphs with fixed Δ is Θ(Δ). In this note we show that in the class of trees with fixed Δ this gap is just Θ(Δ). Namely, we show that if a tree T has the largest vertex degree Δ thenλ1(T)<2Δ−1andμ1(T)<Δ+2Δ−1&les;λ1(G)&les;Δ and Δ+1&les;μ1(G)&les;2Δ.Thus the gap between the maximum and minimum value of λ1(G) and μ1(G) in the class of graphs with fixed Δ is Θ(Δ). In this note we show that in the class of trees with fixed Δ this gap is just Θ(√ of Δ). Namely, we show that if a tree T has the largest vertex degree Δ thenλ1(T)<2Δ−1andμ1(T)<Δ+2Δ−1). Namely, we show that if a tree T has the largest vertex degree Δ thenλ1(T)<2√ of Δ−1andμ1(T)<Δ+2Δ−1 and μ1(T)<Δ+2√ of Δ−1.New bounds are an improvement for Δ&ges;3. [Copyright &y& Elsevier]

The electronic commerce (EC) has been widely studied in the academic as well as practical fields. Especially, a lot of special topics regarding the EC such as B2C and B2B have been investigated in literature. However, there are much less studies about the EC sites themselves. Besides, only a few studies exist about the issues regarding how to adjust the design factors of the EC sites. The main objective of this study is to fill this research void by employing two techniques: (1) cognitive map and (2) linear structural relationship (LISREL). The cognitive map was used to operationalize the causal relationships among design factors of the EC sites, and investigate the simulation to find the optimal strategy of adjusting the design factors. The LISREL was performed to prove the proposed research model, where original Technology Acceptance Model (TAM) [Davis MIS Q. 13 (1989) 319] is adopted as a basic framework for providing causal relationships. Usable questionnaires were collected from 114 respondents who are proved to be qualified for this study. They were educated to surf two typical EC sites appropriately and tested before answering the questionnaires. Those respondents who completed questionnaires successfully were given a book coupon of 5$ equivalent. After LISREL experiments, the proposed research model was tested, and an adjacency matrix was induced which is to be used for the cognitive map simulation. With the adjacency matrix and 15 hypothetical market situations, the cognitive map simulations were successfully performed yielding that the proposed two techniques could be used for successfully adjusting the design factors of the EC sites under consideration in line with the changes in customers'' tastes and market situations. One of the noticeable practical advantages of this study is that decision makers can identify the most relevant design factors and thereby allocate limited resources to them reasonably by performing the cognitive map simulation in advance before doing design adjustment to the EC sites in actuality. [Copyright &y& Elsevier]

The polynomial reconstruction problem (PRP) asks whether for a graph G of order at least 3, the characteristic polynomial can be reconstructed from the p-deck PD(G) of characteristic polynomials of the one-vertex-deleted subgraphs. We show that this is the case for a number of subclasses of the class of graphs with pendant edges. Moreover, we show that if the number of terminal vertices of G is sufficiently high, then G is polynomial reconstructible. [Copyright &y& Elsevier]

In [Linear Algebra Appl. 177 (1992) 137] Smith proved that if H is a Hermitian semidefinite matrix and A is a nonsingular principal submatrix, then the eigenvalues of the Schur complement H/A interlace those of H. In this paper, we refine the latter result and use it to derive eigenvalues interlacing results on an irreducible symmetric nonnegative matrix that involve Perron complements. For an irreducible symmetric nonnegative matrix, we give lower and upper bounds for its spectral radius and also a lower bound for the maximal spectral radius of its principal submatrices of a fixed order. We apply our results to an irreducible symmetric Z-matrix and to the adjacency matrix or the general Laplacian matrix of a connected weighted graph. The equality cases for the bounds for spectral radii or least eigenvalues are also examined. [Copyright &y& Elsevier]

We survey some old and some new characterizations of distance-regular graphs, which depend on information retrieved from their adjacency matrix. In particular, it is shown that a regular graph with d+1 distinct eigenvalues is distance-regular if and only if a numeric equality, involving only the spectrum of the graph and the numbers of vertices at distance d from each vertex, is satisfied. [Copyright &y& Elsevier]

The rapid development of ubiquitous mobile computing has enabled the collection of new types of massive traffic data to understand collective movement patterns in social spaces. Contributing to the understanding of crowd formation and dispersal in populated areas, we developed a model of visitors' dynamic agglomeration patterns at a particular event using dynamic population data. This information, a type of big data, comprised aggregate Global Positioning System (GPS) location data automatically collected from mobile phones without users' intervention over a grid with a spatial resolution of 250 m. Herein, spatial autoregressive models with two-step adjacency matrices are proposed to represent visitors' movement between grids around the event site. We confirmed that the proposed models had a higher goodness-of-fit than those without spatial or temporal autocorrelations. The results also show a significant reduction in accuracy when applied to prediction with estimated values of the endogenous variables of prior time periods. [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]

Abstract: In J.F. Wang and S.N. Shu (2012) [2], the authors lost the condition that the line graph does not contain the cycle of order 4 in Lemma 3.4, which makes their main result (i.e., Theorem 3.1) of that paper not completely correct. Here, some comments are given. [Copyright &y& Elsevier]

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]