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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]