Laplacian spectral characterization of dumbbell graphs and theta graphs

Abstract

Let [Formula: see text] and [Formula: see text] denote the path and cycle on [Formula: see text] vertices, respectively. The dumbbell graph, denoted by [Formula: see text], is the graph obtained from two cycles [Formula: see text], [Formula: see text] and a path [Formula: see text] by identifying each pendant vertex of [Formula: see text] with a vertex of a cycle, respectively. The theta graph, denoted by [Formula: see text], is the graph formed by joining two given vertices via three disjoint paths [Formula: see text], [Formula: see text] and [Formula: see text], respectively. In this paper, we prove that all dumbbell graphs as well as all theta graphs are determined by their [Formula: see text]-spectra.