Which cospectral graphs have same degree sequences

Two graphs are said to be L-cospectral (respectively, Q-cospectral) if they have the same (respectively, signless) Laplacian spectra, and a graph G is said to be L−DS (respectively, Q−DS) if there does not exist other non-isomorphic graph H such that H and G are L-cospectral (respectively, Q-cospectral). Let d1(G)≥d2(G)≥⋯≥dn(G) be the degree sequence of a graph G with n vertices. In this paper, we prove that except for two exceptions (respectively, the graphs with d1(G)∈{4,5}), if H is L-cospectral (respectively, Q-cospectral) with a connected graph G and d2(G)=2, then H has the same degree sequence as G. A spider graph is a unicyclic graph obtained by attaching some paths to a common vertex of the cycle. As an application of our result, we show that every spider graph and its complement graph are both L−DS, which extends the corresponding results of Haemers et al. (2008), Liu et al. (2011), Zhang et al. (2009) and Yu et al. (2014).