Which Q-cospectral graphs have same degree sequences

Abstract

Let Cn denote the cycle with n vertices, and let τ(G) be the number of triangles of G. If G is a graph with n vertices, then μ1(G)≥μ2(G)≥⋯≥μn(G) denote the signless Laplacian spectrum of G. Two graphs are said to be Q-cospectral if they have the same signless Laplacian spectrum. A graph G is said to be Q−DS if there is no other non-isomorphic graph H such that G and H are Q-cospectral. In this paper, we prove that “Let G be a graph with n≥12 vertices and 1≤μn(G)≤μ2(G)≤5<n<μ1(G) or let G be a graph with n≥10 vertices and 0<μn(G)≤μ2(G)≤4<n<μ1(G). If G and H are Q-cospectral, then G and H share the same degree sequences and τ(G)=τ(H)”. As a consequence of our results, we show that all multi-wheel graphs K1∨(Cq1∪Cq2∪⋯∪Cqt) are Q−DS.