We briefly review known results about the signed edge domination number of graphs. In the case of bipartite graphs, the signed edge domination number can be viewed in terms of its bi-adjacency matrix. This motivates the introduction of the signed domination number of a (0, 1)-matrix. We investigate the signed domination number for various classes of (0, 1)-matrices, in particular for regular an...

The spectra of signed matrices have played a fundamental role in social sciences, graph theory, and control theory. In this work, we investigate the computational problems of identifying symmetric signings of matrices with natural spectral properties. Our results are twofold: 1. We show NP-completeness for the following three problems: verifying whether a given matrix has a symmetric signing th...

In this paper simple formulae are derived for calculating the number of spanning trees of different product graphs. The products considered in here consists of Cartesian, strong Cartesian, direct, Lexicographic and double graph. For this purpose, the Laplacian matrices of these product graphs are used. Form some of these products simple formulae are derived and whenever direct formulation was n...

A graph whose edges are labeled either as positive or negative is called a signed graph. In this article, we extend the notion of composition of (unsigned) graphs (also called lexicographic product) to signed graphs. We employ Kronecker product of matrices to express the adjacency matrix of this product of two signed graphs and hence find its eigenvalues when the second graph under composition ...

The enhanced principal rank characteristic sequence (epr-sequence) of an n×n matrix is a sequence `1`2 · · ·`n, where each `k is A, S, or N according as all, some, or none of its principal minors of order k are nonzero. There has been substantial work on epr-sequences of symmetric matrices (especially real symmetric matrices) and real skew-symmetric matrices, and incidental remarks have been ma...

For integers n ≥ 1, k ≥ 0, and k ≤ n, the graph Γn has vertices the 2 vectors of F n 2 and adjacency defined by two vectors being adjacent if they differ in k coordinate positions. In particular Γn is the n-cube, usually denoted by Qn. We examine the binary codes obtained from the adjacency matrices of these graphs when k = 1, 2, 3, following results obtained for the binary codes of the n-cube ...

Let k, l,m, n, and μ be positive integers. A Zμ-scheme of valency (k, l) and order (m,n) is an m × n array (Sij) of subsets Sij ⊆ Zμ such that for each row and column one has ∑n j=1 |Sij | = k and ∑m i=1 |Sij | = l, respectively. Any such scheme is an algebraic equivalent of a (k, l)-semiregular bipartite voltage graph with n and m vertices in the bipartition sets and voltages coming from the c...

This paper is concerned with polynomial optimization problems. We show how to exploit term (or monomial) sparsity of the input polynomials obtain a new converging hierarchy semidefinite programming relaxations. The novelty (and distinguishing feature) such relaxations involve block-diagonal matrices obtained in an iterative procedure performing completion connected components certain adjacency ...

Echo State Networks (ESNs) are Recurrent Neural with fixed input and internal (hidden) weights, adaptable output weights. The hidden part of an ESN can be considered as a discrete-time dynamical system, called reservoir. In classical ESNs, the connections obtained from Erd?s-Rényi graph. A recent study proposed ESNs clustered adjacency matrices (CESNs), where clusters either graphs or Barabási-...

We derive combinatorial necessary conditions for discrete-time quantum walks defined by regular mixed graphs to be periodic. If the walk is periodic, all eigenvalues of time evolution matrices must algebraic integers. Focusing on this, we explore which ring coefficients characteristic polynomials should belong to. On other hand, $\eta$-Hermitian adjacency have implications. From these, can find...