The spectral characterization of butterfly-like graphs

Let a(k)=(a1,a2,…,ak)a(k)=(a1,a2,…,ak) be a sequence of positive integers. A butterfly-like graph  Wp(s);a(k)Wp(s);a(k) is a graph consisting of s  (≥1)(≥1) cycle of lengths p+1p+1, and k  (≥1)(≥1) paths Pa1+1Pa1+1, Pa2+1Pa2+1, …, Pak+1Pak+1 intersecting in a single vertex. The girth of a graph G is the length of a shortest cycle in G. Two graphs are said to be A-cospectral if they have the same adjacency spectrum. For a graph G, if there does not exist another non-isomorphic graph H such that G and H share the same Laplacian (respectively, signless Laplacian) spectrum, then we say that G   is L−DSL−DS (respectively, Q−DSQ−DS). In this paper, we firstly prove that no two non-isomorphic butterfly-like graphs with the same girth are A-cospectral, and then present a new upper and lower bounds for the i  -th largest eigenvalue of L(G)L(G) and Q(G)Q(G), respectively. By applying these new results, we give a positive answer to an open problem in Wen et al. (2015) [17] by proving that all the butterfly-like graphs W2(s);a(k)W2(s);a(k) are both Q−DSQ−DS and L−DSL−DS.