On the computation of Seidel Laplacian eigenvalues for graph-based binary codes

Abstract

Let G=(V,E) be a simple graph with vertex set as V and edge set as E. Within the framework of graph-based binary coding, a graph G is a threshold graph if and only if it can be generated by a binary code of the type 0s11t1…0sk1tk, where si and ti are positive integers with 1≤i≤k. This type of binary code is called a binary generating code of G. In this paper, we address a problem originally posed by Harary and Schwenk in [10] titled “Which graphs have integral spectra?” in which the main focus of our investigations lies in the Seidel Laplacian spectrum of G generated by 0s11t1…0sk1tk. We establish that all eigenvalues of G are integers and multiplicities of eigenvalues are equal to powers of bits 0 and 1 (sizes of strings of 0's and 1's) involved in a binary generating code of G. We exhibit a necessary and sufficient condition in terms of eigenvalues of the matrix associated with a graph-based G binary code to be Seidel Laplacian integral. Furthermore, for any prime p, we explore the Seidel Laplacian spectrum of zero-divisor graphs associated with the ring Zpm (where m is a positive integer) and the polynomial ring Zp[x]/(xp).