The second Zagreb indices of unicyclic graphs with given degree sequences

Let @p=(d"1,d"2,...,d"n) and @p^'=(d"1^',d"2^',...,d"n^') be two different non-increasing degree sequences. We write @[email protected][email protected]^', if and only if @?"i"="1^nd"[email protected]?"i"="1^nd"i^', and @?"i"="1^jd"[email protected][email protected]?"i"="1^jd"i^' for all j=1,2,...,n. Let @C(@p) be the class of connected graphs with degree sequence @p. The second Zagreb index of a graph G is denoted by M"2(G)[email protected]?"u"v"@?"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 @[email protected][email protected]^', @p and @p^' are unicyclic degree sequences and U^* and U^*^* have the maximum second Zagreb indices in @C(@p) and @C(@p^'), respectively, then M"2(U^*)=17 vertices.