Spectral characterizations of dumbbell graphs

Abstract

A dumbbell graph, denoted by Da,b,c, is a bicyclic graph consisting of two vertex- disjoint cycles Ca, Cb and a path Pc+3 (c ≥ -1) joining them having only its end-vertices in common with the two cycles. In this paper, we study the spectral characterization w.r.t. the adjacency spectrum of Da,b,0 (without cycles C4) with gcd(a, b) ≥ 3, and we complete the research started in [J.F. Wang et al., A note on the spectral characterization of dumbbell graphs, Linear Algebra Appl. 431 (2009) 1707-1714]. In particular we show that Da,b,0 with 3≤ gcd(a, b) < a or gcd(a, b) = a and b ≠ 3a is determined by the spectrum. For b = 3a, we determine the unique graph cospectral with Da,3a,0. Furthermore we give the spectral characterization w.r.t. the signless Laplacian spectrum of all dumbbell graphs.