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Abstract Let μ be an eigenvalue of a simple graph G with multiplicity k ≥ 1. A star complement for μ in G is an induced subgraph of G of order n − k with no eigenvalue μ. In this paper, we study the maximal graphs with the star S m as a star complement for −2. The maximal graphs with S 3 , S 4 , S 13 and S 21 as a star complement for −2 are described. We also describe the regular graphs with K 2 , s (s ≥ 2) as a star complement for an eigenvalue μ. [ABSTRACT FROM AUTHOR]

In this paper we develop the concept of energy of a vertex introduced in Arizmendi and Juarez-Romero (2018). We derive basic inequalities, continuity properties, give examples and extend the definition to locally finite graphs. [ABSTRACT FROM AUTHOR]

We study the eigenvalues of the unique connected anti-regular graph A n . Using Chebyshev polynomials of the second kind, we obtain a trigonometric equation whose roots are the eigenvalues and perform elementary analysis to obtain an almost complete characterization of the eigenvalues. In particular, we show that the interval Ω = [ − 1 − 2 2 , − 1 + 2 2 ] contains only the trivial eigenvalues λ = − 1 or λ = 0 , and any closed interval strictly larger than Ω will contain eigenvalues of A n for all n sufficiently large. We also obtain bounds for the maximum and minimum eigenvalues, and for all other eigenvalues we obtain interval bounds that improve as n increases. Moreover, our approach reveals a more complete picture of the bipartite character of the eigenvalues of A n , namely, as n increases the eigenvalues are (approximately) symmetric about the number − 1 2 . We also obtain an asymptotic distribution of the eigenvalues as n → ∞ . Finally, the relationship between the eigenvalues of A n and the eigenvalues of a general threshold graph is discussed. [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 every real α ∈ [ 0 , 1 ] , write A α ( G ) for the matrix A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . This paper presents some extremal results about the spectral radius ρ α ( G ) of A α ( G ) that generalize previous results about ρ 0 ( G ) and ρ 1 / 2 ( G ) . In particular, write B p , q , r be the graph obtained from a complete graph K p by deleting an edge and attaching paths P q and P r to its ends. It is shown that if α ∈ [ 0 , 1 ) and G is a graph of order n and diameter at least k , then ρ α ( G ) ≤ ρ α ( B n − k + 2 , ⌊ k / 2 ⌋ , ⌈ k / 2 ⌉ ) , with equality holding if and only if G = B n − k + 2 , ⌊ k / 2 ⌋ , ⌈ k / 2 ⌉ . This result generalizes results of Hansen and Stevanović [5] , and Liu and Lu [7] . [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]

Let G be a graph with n vertices, and let A ( G ) and D ( G ) denote respectively the adjacency matrix and the degree matrix of G . Define A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) for any real α ∈ [ 0 , 1 ] . The A α -characteristic polynomial of G is defined to be det ( x I n − A α ( G ) ) = ∑ j c α j ( G ) x n − j , where det ( ⁎ ) denotes the determinant of ⁎, and I n is the identity matrix of size n . The A α -spectrum of G consists of all roots of the A α -characteristic polynomial of G . A graph G is said to be determined by its A α -spectrum if all graphs having the same A α -spectrum as G are isomorphic to G . In this paper, we first formulate the first four coefficients c α 0 ( G ) , c α 1 ( G ) , c α 2 ( G ) and c α 3 ( G ) of the A α -characteristic polynomial of G . And then, we observe that A α -spectra are much efficient for us to distinguish graphs, by enumerating the A α -characteristic polynomials for all graphs on at most 10 vertices. To verify this observation, we characterize some graphs determined by their A α -spectra. [ABSTRACT FROM AUTHOR]

A signed graph G σ consists of an underlying graph G and a sign function σ , which assigns each edge uv of G a sign σ ( u v ) , either positive or negative. The adjacency matrix of G σ is defined as A ( G σ ) = ( a u , v σ ) with a u , v σ = σ ( u v ) a u , v , where a u , v = 1 if u , v ∈ V ( G ) are adjacent, and a u , v = 0 otherwise. The positive inertia index of G σ , written as p ( G σ ) , is defined to be the number of positive eigenvalues of A ( G σ ) . Recently, Yu et al. (2016) [12] characterized the signed graphs G σ with pendant vertices such that p ( G σ ) = 2 . In this paper, we extend the above work to a more general case, characterizing the signed graphs G σ with cut points whose positive inertia index is 2. [ABSTRACT FROM AUTHOR]

Let G be a complex unit gain graph which is obtained from an undirected graph Γ by assigning a complex unit φ ( v i v j ) to each oriented edge v i v j such that φ ( v i v j ) φ ( v j v i ) = 1 for all edges. The Laplacian matrix of G is defined as L ( G ) = D ( G ) − A ( G ) , where D ( G ) is the degree diagonal matrix of Γ and A ( G ) = ( a i j ) has a i j = φ ( v i v j ) if v i is adjacent to v j and a i j = 0 otherwise. In this paper, we provide a combinatorial description of det ( L ( G ) ) that generalizes that for the determinant of the Laplacian matrix of a signed graph. [ABSTRACT FROM AUTHOR]

Kautz digraphs K ( d , ℓ ) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related with these, the cyclic Kautz C K ( d , ℓ ) and the subKautz s K ( d , 2 ) digraphs were recently introduced by Böhmová, Huemer and the author. In this paper we propose a new method to obtain the complete spectra of subKautz s K ( d , 2 ) and cyclic Kautz C K ( d , 3 ) digraphs, for all d ≥ 3 , through the Hoffman–McAndrew polynomial and regular partitions. This approach can be useful to find the spectra of other families of digraphs with high regularity. [ABSTRACT FROM AUTHOR]

We consider only simple graphs. A graph G is said to be nonsingular if its adjacency matrix A ( G ) is nonsingular. The inverse of a nonsingular graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A ( G ) via a diagonal matrix of ±1s. An invertible graph G is said to be a self-inverse graph if G is isomorphic to its inverse. Let H be the class of connected bipartite graphs with unique perfect matchings. We consider the problem of characterizing the graphs in H which are self-inverse graphs. In this article, we answer this problem for the class of unicyclic graphs. The study of the strong reciprocal eigenvalue property is closely related to the study of graphs which are isomorphic to their inverse graphs. A nonsingular graph G is said to satisfy the property (SR) if the reciprocal of each eigenvalue of the adjacency matrix A ( G ) is also an eigenvalue of A ( G ) and they both have the same multiplicities. In this article, we show that if a unicyclic graph G in H has property (SR), then G is invertible and the inverse graph of G is unicyclic. As an application, we show that a noncorona unicyclic graph in H with property (SR) can have one of the five specified structures. Finally, our discussions lead to the following problem. Does there exist a unicyclic graph G ∈ H which has property (SR) but G is not self-inverse? [ABSTRACT FROM AUTHOR]

We consider only simple graphs. Consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contracting all matching edges is also bipartite. On the class H g of such graphs G the equivalence of the following statements are known. i) The reciprocal of the spectral radius of the adjacency matrix A ( G ) is the least positive eigenvalue of the adjacency matrix. ii) The graph is isomorphic to its inverse, where the inverse of a graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A ( G ) via a diagonal matrix of ±1s. iii) The graph has the reciprocal eigenvalue property, that is, the reciprocal of each eigenvalue of the adjacency matrix A ( G ) is also an eigenvalue of A ( G ) . iv) The graph has the strong reciprocal eigenvalue property, that is, the reciprocal of each eigenvalue of the adjacency matrix A ( G ) is also an eigenvalue of A ( G ) and they both have the same multiplicities. v) The graph is a corona graph, that is, it is obtained by taking a bipartite graph and then by inserting a new adjacent vertex of degree one at each vertex. It is known that the statements iii) and iv) are not equivalent, if we allow nonbipartite graphs. However, the equivalence of these two is not yet known for the class of bipartite graphs with unique perfect matchings, where we relax the condition of bipartiteness of the contracted graph. In this article, we establish the equivalence of the first four statements for a class larger than H g . We then supply a constructive structural description of those graphs. [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 every real α ∈ [ 0 , 1 ] , write A α ( G ) for the matrix A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . Let α 0 ( G ) be the smallest α for which A α ( G ) is positive semidefinite. It is known that α 0 ( G ) ≤ 1 / 2 . The main results of this paper are: (1) if G is d -regular then α 0 = − λ min ( A ( G ) ) d − λ min ( A ( G ) ) , where λ min ( A ( G ) ) is the smallest eigenvalue of A ( G ) ; (2) G contains a bipartite component if and only if α 0 ( G ) = 1 / 2 ; (3) if G is r -colorable, then α 0 ( G ) ≥ 1 / r . [ABSTRACT FROM AUTHOR]

Strona and Veech (2015) developed a new node segregation (or node overlap) index for analysing ecological network structure based on the Veech (2013)’s species co-occurrence probabilistic model, which was originally applied to species-site matrices. However, a species-site matrix for analysing species co-occurrence patterns and an adjacency matrix for characterising unimode network structures are different. Directly applying Veech’s species co-occurrence probabilistic model to adjacency matrices in unimode food webs is problematic. The central critical problem is related to the number of free species (or nodes/vertices) in the unimode network that can be the neighbors (have links to connect) of a focused species or a focused pair of species. This number is typically less than the total number of species in real food webs. That is, species are not independent from each other in unimode networks. For a simple undirected unimode network without self-loops, based on the criterion whether there is a link between two species for a focused pair, a correct probabilistic model is developed to accurately compute the probability of observing some shared neighbors for a pair of species in the network. Numerical simulation show that the node overlap calculated using the correct and original probabilistic models present remarkable differences, especially when a unimode network is nested and contains generalists. In summary, The correct probabilistic model should be used if ones want Strona and Veech (2015)’s node segregation index to work for unimode food webs. [ABSTRACT FROM AUTHOR]

Many networks exhibit small-world properties. The structure of a small-world network is characterized by short average path lengths and high clustering coefficients. Few graph layout methods capture this structure well which limits their effectiveness and the utility of the visualization itself. Here we present an extension to our novel graphTPP layout method for laying out small-world networks using only their topological properties rather than their node attributes. The Watts–Strogatz model is used to generate a variety of graphs with a small-world network structure. Community detection algorithms are used to generate six different clusterings of the data. These clusterings, the adjacency matrix and edgelist are loaded into graphTPP and, through user interaction combined with linear projections of the adjacency matrix, graphTPP is able to produce a layout which visually separates these clusters. These layouts are compared to the layouts of two force-based techniques. graphTPP is able to clearly separate each of the communities into a spatially distinct area and the edge relationships between the clusters show the strength of their relationship. As a secondary contribution, an edge-grouping algorithm for graphTPP is demonstrated as a means to reduce visual clutter in the layout and reinforce the display of the strength of the relationship between two communities. [ABSTRACT FROM AUTHOR]

The n -vertex graph G ( = Γ ( G ) ) with a non-singular real symmetric adjacency matrix G , having a zero diagonal and singular ( n − 1 ) × ( n − 1 ) principal submatrices is termed a NSSD, a Non-Singular graph with a Singular Deck. NSSDs arose in the study of the polynomial reconstruction problem and were later found to characterise non-singular molecular graphs that are distinct omni-conductors and ipso omni-insulators. Since both matrices G and G − 1 represent NSSDs Γ ( G ) and Γ ( G − 1 ) , the value of the nullity of a one-, two- and three-vertex deleted subgraph of G is shown to be determined by the corresponding subgraph in Γ ( G − 1 ) . Constructions of infinite subfamilies of non-NSSDs are presented. NSSDs with all two-vertex deleted subgraphs having a common value of the nullity are referred to as G -nutful graphs. We show that their minimum vertex degree is at least 4. [ABSTRACT FROM AUTHOR]

The energy of a simple graph G , E ( G ) , is the sum of the absolute values of the eigenvalues of its adjacency matrix. The energy of line graph and the signless Laplacian energy of graph G are denoted by E ( L G ) ( L G is the line graph of G ) and LE + ( G ) , respectively. In this paper we obtain a relation between E ( L G ) and LE + ( G ) of graph G . From this relation we characterize all the graphs satisfying E ( L G ) = LE + ( G ) + 4 m n − 4 . We also present a relation between E ( G ) and E ( L G ) . Moreover, we give an upper bound on E ( L G ) of graph G and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

Given a graph G . The fewest number of vertices whose deletion yields a bipartite graph from G was defined by S. Fallat and Yi-Zheng Fan to be the vertex bipartiteness of G and it is denoted by υ b ( G ) . We consider the set Σ k ( n ) defined by { G = ( V ( G ) , E ( G ) ) : G connected , | V ( G ) | = n and υ b ( G ) ≤ k } . In this work we identify the graph in Σ k ( n ) with maximum spectral radius and maximum signless Laplacian spectral radius. [ABSTRACT FROM AUTHOR]

We apply matrix theory over F 2 to understand the nature of so-called “successful pressing sequences” of black-and-white vertex-colored graphs. These sequences arise in computational phylogenetics, where, by a celebrated result of Hannenhalli and Pevzner, the space of sortings-by-reversal of a signed permutation can be described by pressing sequences. In particular, we offer several alternative linear-algebraic and graph-theoretic characterizations of successful pressing sequences, describe the relation between such sequences, and provide bounds on the number of them. We also offer several open problems that arose as a result of the present work. [ABSTRACT FROM AUTHOR]

A finite sequence d: d 1 , d 2 , d 3 ,. .. ., d n of nonnegative integers is said to be graphical if there exists some finite simple graph G, having vertex set V={ v 1 , v 2 , v 3 , …., v n } such that each v i has degree d i (1 ≤ i ≤ n). In this paper we have proposed an algorithm that takes a non-increasing sequence as input and determines whether the given degree sequence is graphic by constructing the adjacency matrix from the given sequence in non-increasing order and checking whether it can be orthogonally diagonalizable. The matrix generated in this process can be used to determine a lot of interesting information regarding the characteristic of the graph directly from the given degree sequence. [ABSTRACT FROM AUTHOR]

In this paper the unique graph with maximal index (i.e. the largest eigenvalue of the adjacency matrix) is identified among all connected bipartite graphs of order n and size n + k , under the assumption that k ≥ 0 and n ≥ k + 5 . [ABSTRACT FROM AUTHOR]

Let Γ = ( G , σ ) be a signed graph, where G is the underlying simple graph and σ : E ( G ) → { + , − } is the sign function on the edges of G . In this paper we consider the spectral characterization problem extended to the adjacency matrix and Laplacian matrix of signed graphs. After giving some basic results, we study the spectral determination of signed lollipop graphs, and we show that any signed lollipop graph is determined by the spectrum of its Laplacian matrix. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with the adjacency matrix A ( G ) . A well-known result of Cvetković and Gutman states that removing a pendant vertex and its neighbour, does not change the nullity of the graph. We generalize this theorem and some other theorems similar to it for an arbitrary eigenvalue of a graph. Also, for an arbitrary eigenvalue λ , we use a corresponding star set to delete some subgraphs and to determine the multiplicity of λ . We use star sets to find some Parter–Wiener vertices, in trees. We state our results for real symmetric matrices (and so for weighted graphs) and as a corollary we use them for simple graphs. Furthermore, we use these methods to obtain some results for λ = 0 , ± 1 . [ABSTRACT FROM AUTHOR]

For a fixed real number r we give several necessary and/or sufficient conditions for a graph to have the second largest eigenvalue of the adjacency matrix, or signless Laplacian matrix, less then or equal to r . [ABSTRACT FROM AUTHOR]

Assuming a uniform random model of selecting creation sequences, we show that almost every connected threshold graph has more negative than positive eigenvalues. We show that no threshold graphs have eigenvalues in ( − 1 , 0 ) . Sequences of equienergetic graphs are given including the striking result that for all n ≥ 3 , there exist n − 1 threshold graphs of order n 2 , pairwise noncospectral, each equienergetic to K n 2 . For all n > 8 , we find an n -vertex hyperenergetic threshold graph. [ABSTRACT FROM AUTHOR]

Let G be a graph with adjacency matrix A, and let H(t)=exp(itA). For an eigenvalue μ of A with multiplicity k, a star set for μ in G is a vertex set X of G such that |X|=k and the induced subgraph G-X does not have μ as an eigenvalue. G is said to have perfect state transfer from the vertex u to the vertex v if there is a time τ such that |H(τ)u,v|=1. The unitary operator H(t) has important applications in the transfer of quantum information. In this paper, we give an expression of H(t). For a star set X of graph G, perfect state transfer does not occur between any two vertices in X. We also give some results for the existence of perfect state transfer in a graph. [ABSTRACT FROM AUTHOR]

We unify and generalize several inequalities for the number wk of walks of length k in graphs, and for the entry sum of matrix powers. First, we present a weighted sandwich theorem for Hermitian matrices which generalizes a matrix theorem by Marcus and Newman and which further generalizes our former unification of inequalities for the number of walks in undirected graphs by Lagarias et al. and by Dress and Gutman. The new inequality uses an arbitrary nonnegative weighting of the indices (vertices) which allows to apply the theorem to index (vertex) subsets (i.e., inequalities considering the number wk(S,S) of walks of length k that start at a vertex of a given vertex subset S and that end within the same subset). We also deduce a stronger variation of the sandwich theorem for the case of positive-semidefinite Hermitian matrices which generalizes another inequality of Marcus and Newman. Further, we show a similar theorem for nonnegative symmetric matrices which is another unification and generalization of inequalities for walk numbers by Erdős and Simonovits, by Dress and Gutman, and by Ilić and Stevanović. In the last part, we generalize lower bounds for the spectral radius of adjacency matrices and upper bounds for the energy of graphs. [ABSTRACT FROM AUTHOR]

Abstract: Let be a weighted graph. The inertia of is the triple , where , , are the numbers of the positive, negative and zero eigenvalues of the adjacency matrix of including their multiplicities, respectively. , are called the positive, negative indices of inertia of , respectively. In this paper we present a lower bound for the positive, negative indices of weighted unicyclic graphs of order n with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound. Moreover, we characterize the weighted unicyclic graphs of order n with two positive, two negative and at least zero eigenvalues, respectively. [Copyright &y& Elsevier]

Abstract: We calculate the Smith normal form of the adjacency matrix of each of the following graphs or their complements (or both): complete graph, cycle graph, square of the cycle, power graph of the cycle, distance matrix graph of cycle, Andrásfai graph, Doob graph, cocktail party graph, crown graph, prism graph, Möbius ladder. The proofs operate by finding the abelianization of a cyclically presented group whose relation matrix is column equivalent to the required adjacency matrix. [Copyright &y& Elsevier]

Abstract: Let G be a graph with vertex set , and let H be the graph obtained by attaching one pendant path of length at vertex . For a real symmetric matrix A whose graph is H, let denote the multiplicity of an eigenvalue μ of A. From a result in da Fonseca (2005) [7], we know that . In this note, we characterize the case . We also give two upper bounds on eigenvalue multiplicity of trees and unicyclic graphs, which are generalizations of some results in Rowlinson (2010) [10]. [Copyright &y& Elsevier]

Abstract: A graph is called reflexive if its second largest eigenvalue does not exceed 2. In this paper, we determine all reflexive bipartite regular graphs. Any bipartite regular graph of degree at most 2 is reflexive as well as its bipartite complement. Apart from them, there is a finite number of resulting graphs. [Copyright &y& Elsevier]

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]

Abstract: We give upper and lower bounds for the spectral radius of a nonnegative matrix using its row sums and characterize the equality cases if the matrix is irreducible. Then we apply these bounds to various matrices associated with a graph, including the adjacency matrix, the signless Laplacian matrix, the distance matrix, the distance signless Laplacian matrix, and the reciprocal distance matrix. Some known results in the literature are generalized and improved. [Copyright &y& Elsevier]

Abstract: Let G be a threshold graph of order n with adjacency matrix A. We present an algorithm for constructing a diagonal matrix congruent to for any x. An application, using Sylvesterʼs Law of Inertia, can determine how many eigenvalues lie in an interval, allowing efficient approximation. We prove that eigenvalues of threshold graphs, other than 0 or −1, are simple. We give the spectrum for threshold graphs , represented by . When , has the minimum eigenvalue among threshold graphs of order n, and a formula for is given. There is one more graph if . [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: The energy of a graph G is the sum of the singular values of its adjacency matrix. It is a graph invariant used in mathematical chemistry. The normalized incidence energy of the graph G, denoted by , is defined as the sum of the singular values of its normalized incidence matrix. We find some upper and lower bounds and determine the Coulson integral formula for . Based on the integral formula, we give a way to compare the normalized incidence energies. Moreover we show a relation between normalized incidence energy and Randić energy. [Copyright &y& Elsevier]

Abstract: Consider a graph whose vertices are the distinguishable submultisets of a finite multiset such that two vertices and are adjacent if and only if . In this paper, the walks on these graphs are linked to square and domino-tilings of rectangular and circular boards. Among the results provided, the number of walks of any given length is shown to be a product of generalized Fibonacci numbers. As an application, the eigenvalues are computed for the (unrestricted) zero-divisor graphs of finite commutative rings without nonzero nilpotents. [Copyright &y& Elsevier]

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]

Abstract: Let G be a simple connected graph of order n with degree sequence in non-increasing order. The spectral radius of G is the largest eigenvalue of its adjacency matrix. For each positive integer at most n, we give a sharp upper bound for by a function of which generalizes a series of previous results. [Copyright &y& Elsevier]

Abstract: We derive some structural and spectral properties of regular bipartite graphs with three distinct non-negative eigenvalues. Next, we consider the relations between these graphs and two-class partially balanced incomplete block designs, and we present a number of situations when the graphs we consider are in fact the incidence graphs of those designs. As a consequence, we give a several constructions of connected regular bipartite graphs with six distinct eigenvalues, and we also determine all such graphs with degree 3, and all such graphs on at most 20 vertices. [Copyright &y& Elsevier]

Abstract: A generalized H-join operation of a family of graphs , where H has order p, constrained by a family of vertex subsets , , is introduced. When each vertex subset is -regular, it is deduced that all non-main adjacency eigenvalues of , different from , remain as eigenvalues of the graph G obtained by this operation. If each is -regular and all the vertex subsets are such that , the H-generalized join constrained by these vertex subsets coincides with the H-join operation. Furthermore, some applications on the spread of graphs are presented. [Copyright &y& Elsevier]

Abstract: Let be a connected graph with vertex set and a weight function that assigns a nonnegative number to each of its vertices. Then, the -moment of at vertex is defined to be , where stands for the distance function. Adding up all these numbers, we obtain the -moment of : This parameter generalizes, or it is closely related to, some well-known graph invariants, such as the Wiener index , when for every , and the degree distance , obtained when , the degree of vertex . In this paper we derive some exact formulas for computing the -moment of a graph obtained by a general operation called graft product, which can be seen as a generalization of the hierarchical product, in terms of the corresponding -moments of its factors. As a consequence, we provide a method for obtaining nonisomorphic graphs with the same -moment for every (and hence with equal mean distance, Wiener index, degree distance, etc.). In the case when the factors are trees and/or cycles, techniques from linear algebra allow us to give formulas for the degree distance of their product. [Copyright &y& Elsevier]

Abstract: The positive index of inertia of a graph G, denoted by , is the number of the positive eigenvalues of the adjacency matrix of G. In this paper, we investigate the minimal positive index of inertia among all bicyclic graphs of order n with pendant vertices, and characterize the bicyclic graphs with positive index 1 or 2 among all bicyclic graphs of order n. [Copyright &y& Elsevier]

Abstract: For many types of graphs, criteria have been discovered that give necessary and sufficient conditions for an integer sequence to be the degree sequence of such a graph. These criteria tend to take the form of a set of inequalities, and in the case of the Erdős–Gallai criterion (for simple undirected graphs) and the Gale–Ryser criterion (for bipartite graphs), it has been shown that the number of inequalities that must be checked can be reduced significantly. We show that similar reductions hold for the corresponding criteria for many other types of graphs, including bipartite -multigraphs, bipartite graphs with structural edges, directed graphs, -multigraphs, and tournaments. We also prove a reduction for imbalance sequences. [Copyright &y& Elsevier]

Abstract: Let be a simple graph of order with triangle(s). Also let be the eigenvalues of the adjacency matrix of graph . X. Yong [X. Yong, On the distribution of eigenvalues of a simple undirected graph, Linear Algebra Appl. 295 (1999) 73–80] conjectured that (i) is complete if and only if and also (ii) is complete if and only if . Here we disprove this conjecture by a counter example. Wang et al. [J.F. Wang, F. Belardo, Q.X. Huang, B. Borovićanin, On the two largest Q-eigenvalues of graphs, Discrete Math. 310 (2010) 2858–2866] conjectured that friendship graph is determined by its adjacency spectrum. Here we prove this conjecture. The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity of a graph is the mean value of eccentricities of all vertices of . Moreover, we mention three conjectures, obtained by the system AutoGraphiX, about the average eccentricity , girth and the spectral radius of graphs (see Aouchiche (2006) [1], available online at http://www.gerad.ca/~agx/). We give a proof of one conjecture and disprove two conjectures by counter examples. [Copyright &y& Elsevier]

Abstract: The rank of a graph is that of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced trees as well as bipartite graphs with a given rank and characterize those graphs achieving the maximum order. [Copyright &y& Elsevier]

Abstract: In this paper we introduce a new parameter for a graph called the minimum universal rank. This parameter is similar to the minimum rank of a graph. For a graph G the minimum universal rank of G is the minimum rank over all matrices of the form where A is the adjacency matrix of G, J is the all ones matrix and D is the matrix with the degrees of the vertices in the main diagonal, and are scalars. Bounds for general graphs based on known graph parameters are given, as is a formula for the minimum universal rank for regular graphs based on the multiplicity of the eigenvalues of A. The exact value of the minimum universal rank of some families of graphs are determined, including complete graphs, complete bipartite graph, paths and cycles. Bounds on the minimum universal rank of a graph obtained by deleting a single vertex are established. It is shown that the minimum universal rank is not monotone on induced subgraphs, but bounds based on certain induced subgraphs, including bounds on the union of two graphs, are given. [Copyright &y& Elsevier]

Abstract: Some graphs Γ have the following property : the configuration graph (i.e. the non–collinearity graph) of the neighbourhood geometry of Γis isomorphic to Γ. For instance, the ubiquitous Petersen graph satisfies . The purpose of this paper is to reveal repercussions of property on adjacency matrices A for Γ. This will be achieved in terms of invariance under (powers of) the following mapping Θ: denote by I and J the identity matrix and the all one matrix, respectively, both of order n = k 2+1, and define Θ(A) := (k−1)I + J−A 2. If k =3 and A is an adjacency matrix for the Petersen graph, it is well known that Θ(A)= A. In 1960, Hoffman and Singleton showed that for arbitrary k the matrix equation Θ(A)= A has only very few solutions, namely for k =2,3,7 and possibly for k =57. We prove that the property implies the existence of an integer m ⩾1 such that Θ m (A)= A. We determine a standard form for such matrices which is invariant under the action of Θ. In particular, we characterize all solutions, on A, to Θ m (A)= A for k =3 and k =4, where A is an adjacency matrix of some k–regular graph, solving a conjecture posed by the authors. [Copyright &y& Elsevier]

Abstract: We determine all trees whose second largest eigenvalue does not exceed . Next, we consider two classes of bipartite graphs, regular and semiregular, with small number of distinct eigenvalues. For all graphs considered we determine those whose second largest eigenvalue is equal to . Some additional results are also given. [Copyright &y& Elsevier]

Abstract: Let be two simple connected graphs. Denote the corona and the edge corona of by and , respectively. In this paper, we first introduce a new invariant, the -coronal of a graph matrix , where the matrix is associated with a graph in a prescribed way. Then it is used to compute the signless Laplacian spectrum of and in terms of the signless Laplacian spectrum of and . In addition, the spectrum of are also given in terms of the spectrum of and . Finally, as an application of these results, we construct many pairs of nonisomorphic signless Laplacian cospectral graphs. [Copyright &y& Elsevier]