ISI-Equienergetic Graphs

Abstract

The ISI-energy εisi(G) of a graph G=(V,E) is the sum of the absolute values of the eigenvalues of the ISI-matrix C(G)=[cij]n×n in which cij=d(vi)d(vj)d(vi)+d(vj) if vivj∈E(G) and cij=0 otherwise. d(vi) denotes the degree of vertex vi∈V. As a class of graph energy, ISI-energy can be utilized to ascertain the general energy of conjugated carbon molecules. Two non-isomorphic graphs of the same order are said to be ISI-equienergetic if their ISI-energies are equal. In this paper, we construct pairs of connected, ISI-noncospectral, ISI-equienergetic graphs of order n for all n≥9. In addition, for n-vertex r(r≥3)-regular graph G, and for each k≥2, we obtain εisi(Lk(G)¯), depending only on n and r. This result enables a systematic construction of pairs of ISI-noncospectral graphs of the same order, having equal ISI-energies.