The cospectral equivalence classes of graphs having an isolated vertex

Abstract: Let be the characteristic polynomial of a graph . Two graphs and are cospectral, denoted by , if . By we denote the cospectral equivalence class determined by under “”. A graph is said to be determined by its spectrum (or simply is a DS-graph) if whenever . In this paper, we determine the cospectral equivalence classes of three kinds of graphs having an isolated vertex, find several DS-graphs and identify the graph that has the fourth minimum index among all connected graphs with vertices. [Copyright &y& Elsevier]