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It is a well-known fact that a graph of diameter d has at least d + 1 eigenvalues. A graph is d -extremal, if it has diameter d and exactly d + 1 eigenvalues. A graph is split if its vertex set can be partitioned into a clique and a stable set. Such graphs have diameter at most 3. We obtain a complete classification of the connected bidegreed 3-extremal split graphs using the association of split graphs with combinatorial designs. We also construct certain families of non-bidegreed 3-extremal split graphs. [ABSTRACT FROM AUTHOR]

In this paper we consider a concept of strong regularity of signed graphs—a generalization of strong regularity of graphs. We establish certain structural and spectral properties of such signed graphs and determine or characterize those belonging to specified classes. Theoretical results are followed by examples and discussions. [ABSTRACT FROM AUTHOR]

The pineapple graph K p q is obtained by appending q pendant edges to a vertex of a complete graph K p (p ≥ 3 , q ≥ 1). We prove that among connected graphs, the pineapple graph is determined by its adjacency spectrum. Moreover, we determine all disconnected graphs which are cospectral with a pineapple graph. Thus we find for which values of p and q the pineapple graph K p q is determined by its adjacency spectrum. The main tool is a recent classification of all graphs with all but three eigenvalues equal to 0 or − 1 by the third author. [ABSTRACT FROM AUTHOR]

We present some applications of a new matrix approach for studying the properties of the lift Γ α of a voltage digraph, which has arcs weighted by the elements of a group. As a main result, when the involved group is Abelian, we completely determine the spectrum of Γ α. As some examples of our technique, we study some basic properties of the Alegre digraph, and completely characterize the spectrum of a new family of digraphs, which contains the generalized Petersen graphs, and the Hoffman–Singleton graph. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph with n vertices and m edges. Let W (G) = (G , w) be the weighted graph corresponding to G. Let λ 1 , λ 2 , ... , λ n be the eigenvalues of the adjacency matrix A (W (G)) of the weighted graph W (G). The energy E (W (G)) of a weighted graph W (G) is defined as the sum of absolute value of the eigenvalues of W (G). In this paper, we obtain upper bounds for the energy E (W (G)) , in terms of the sum of the squares of weights of the edges, the maximum weight, the maximum degree d 1 , the second maximum degree d 2 and the vertex covering number τ of the underlying graph G. As applications to these upper bounds we obtain some upper bounds for the energy (adjacency energy), the extended graph energy, the Randić energy and the signed energy of the connected graph G. We also obtain some new families of weighted graphs where the energy increases with increase in weights of the edges. [ABSTRACT FROM AUTHOR]

Let G = (V (G) , E (G)) be a simple graph with vertex set V (G) and edge set E (G) , c (G) = | E (G) | − | V (G) | + θ (G) be the dimension of cycle spaces of G with θ (G) the number of connected components of G , m (G) be the matching number of G , and η (G) be the nullity of G. It was shown in Wang and Wong (2014) that the nullity η (G) of G is bounded by an upper bound and a lower bound as | V (G) | − 2 m (G) − c (G) ≤ η (G) ≤ | V (G) | − 2 m (G) + 2 c (G). Graphs with nullity attaining the upper bound and the lower bound have been characterized by Song et al. (2015) and Wang (2016), respectively. In this paper, we prove that there is no graph with nullity η (G) = | V (G) | − 2 m (G) + 2 c (G) − 1. For fixed c (G) , infinitely many connected graphs with nullity | V (G) | − 2 m (G) + 2 c (G) − k , (0 ≤ k ≤ 3 c (G) , k ≠ 1) are also constructed. [ABSTRACT FROM AUTHOR]

Abstract We investigate two Bruhat (partial) orders on graphs with vertices labeled 1 , 2 , ... , n and with a specified degree sequence R , equivalently, symmetric (0 , 1) -matrices with zero trace and a specified row sum vector R (adjacency matrices of such graphs). One is motivated by the classical Bruhat order on permutations while the other one, more restrictive, is defined by a switch of a pair of disjoint edges. In the Bruhat order, one seeks to concentrate the edges of a graph with a given degree sequence among the vertices with smallest labels, thereby producing a minimal graph in this order. We begin with a discussion of graphs whose isomorphism class does not change under a switch. Then we are interested in when the two Bruhat orders are identical. For labeled graphs of regular degree k , we show that the two orders are identical for k ≤ 2 but not for k = 3. [ABSTRACT FROM AUTHOR]

Abstract In this paper we introduce a new graph matrix, named the anti-adjacency matrix or eccentricity matrix, which is constructed from the distance matrix of a graph by keeping for each row and each column only the largest distances. This matrix can be interpreted as the opposite of the adjacency matrix, which is instead constructed from the distance matrix of a graph by keeping for each row and each column only the distances equal to 1. We show that the eccentricity matrix of trees is irreducible, and we investigate the relations between the eigenvalues of the adjacency and eccentricity matrices. Finally, we give some applications of this new matrix in terms of molecular descriptors, and we conclude by proposing some further research problems. [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]

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]

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 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: A new operation on graphs is introduced and some of its properties are studied. We call it hierarchical product, because of the strong (connectedness) hierarchy of the vertices in the resulting graphs. In fact, the obtained graphs turn out to be subgraphs of the cartesian product of the corresponding factors. Some well-known properties of the cartesian product, such as reduced mean distance and diameter, simple routing algorithms and some optimal communication protocols are inherited by the hierarchical product. We also address the study of some algebraic properties of the hierarchical product of two or more graphs. In particular, the spectrum of the binary hypertree (which is the hierarchical product of several copies of the complete graph on two vertices) is fully characterized; turning out to be an interesting example of graph with all its eigenvalues distinct. Finally, some natural generalizations of the hierarchic product are proposed. [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]

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]

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]