Abstract
Abstract Let A and S be the adjacency and the Seidel matrix of a graph G respectively. A-energy is the ordinary energy E(G) of a graph G defined as the sum of the absolute values of eigenvalues of A. Analogously, S-energy is the Seidel energy ES(G) of a graph G defined to be the sum of the absolute values of eigenvalues of the Seidel matrix S. In this article, certain class of A-equienergetic and S-equienergetic graphs are presented. Also some linear relations on A-energies and S-energies are given.
Highlights
Seidel introduced real symmetric {0, ±1}matrix called the Seidel matrix S is defined as S = J − I − 2A, where J is the matrix of order n whose all entries are equal to 1 and I is the identity matrix of order n
If λ1, λ2, . . . , λk are the distinct S-eigenvalues of G of order n with respective multiplicities m1, m2, . . . , mk, the Seidel spectrum or S-spectrum of G is denoted by
The Cartesian product of two graphs G1 and G2 is the graph G1 G2 with vertex set V(G1) × V(G2), in which the vertices (u1, u2) and (v1, v2) are adjacent if either u1 is adjacent to v1 in G1 and u2 is equal to v2 or u1 is equal to v1 and u2 is adjacent to v2 in G2
Summary
Two graphs G1 and G2 of same order are said to be equienergetic or Aequienergetic if E(G1) = E(G2). Ramane et al [21] obtained non-cospectral A-equienergetic iterated line graphs from regular graphs. One of the interesting and difficult problem in the study of energy of a graph in spectral graph theory is to find non-isomorphic graphs of same order with same energy. For in the literature the linear relations on energies of two non isomorphic graphs are not well studied except A-equienergetic or S-equienergetic graphs.
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