The second Zagreb indices of unicyclic graphs with given degree sequences

Let π=(d1,d2,…,dn)π=(d1,d2,…,dn) and π′=(d1′,d2′,…,dn′) be two different non-increasing degree sequences. We write π◃π′π◃π′, if and only if ∑i=1ndi=∑i=1ndi′, and ∑i=1jdi≤∑i=1jdi′ for all j=1,2,…,nj=1,2,…,n. Let Γ(π)Γ(π) be the class of connected graphs with degree sequence ππ. The second Zagreb index of a graph GG is denoted by M2(G)=∑uv∈E(G)d(u)d(v)M2(G)=∑uv∈E(G)d(u)d(v). In this paper, we characterize an extremal unicyclic graph that achieves the maximum second Zagreb index in the class of unicyclic graphs with given degree sequence, and we also prove that if π◃π′π◃π′, ππ and π′π′ are unicyclic degree sequences and U∗U∗ and U∗∗U∗∗ have the maximum second Zagreb indices in Γ(π)Γ(π) and Γ(π′)Γ(π′), respectively, then M2(U∗)