A connected graph G=(V(G),E(G)) is called a quasi-unicyclic graph,if there exists u0∈V(G) such that G-u0 is a unicyclic graph.Denote Q(n,d0)={G:G is a quasi-unicyclic graph of order n with G-u0 being a unicyclic grap...A connected graph G=(V(G),E(G)) is called a quasi-unicyclic graph,if there exists u0∈V(G) such that G-u0 is a unicyclic graph.Denote Q(n,d0)={G:G is a quasi-unicyclic graph of order n with G-u0 being a unicyclic graph and dG(u0)=d0}.Let A(G) be the adjacency matrix of a graph G,and let λ1(G),λ2(G),…,λn(G) be the eigenvalues in non-increasing order of A(G).The number n∑i=1λi^k(k=0,1,…,n-1) is called the k-th spectral moment of G,denoted by Sk(G).Let S(G)=(S0(G),S1(G),…,Sn-1(G)) be the sequence of spectral moments of G.For two graphs G1,G2,we have■ if for some k(k=1,2,…,n-1), and we have Si(G1)=Si(G2)(i=0,1,…,k-1) and Sk(G1)<Sk(G2).In this paper,we determine the second to the fourth largest quasi-unicyclic graphs,in an S-order,in the set Q(n,d0),respectively.展开更多
基金Supported by China Scholarship Council(201808420093)Research Project of Education Bureau of Wuhan(2014017)
文摘A connected graph G=(V(G),E(G)) is called a quasi-unicyclic graph,if there exists u0∈V(G) such that G-u0 is a unicyclic graph.Denote Q(n,d0)={G:G is a quasi-unicyclic graph of order n with G-u0 being a unicyclic graph and dG(u0)=d0}.Let A(G) be the adjacency matrix of a graph G,and let λ1(G),λ2(G),…,λn(G) be the eigenvalues in non-increasing order of A(G).The number n∑i=1λi^k(k=0,1,…,n-1) is called the k-th spectral moment of G,denoted by Sk(G).Let S(G)=(S0(G),S1(G),…,Sn-1(G)) be the sequence of spectral moments of G.For two graphs G1,G2,we have■ if for some k(k=1,2,…,n-1), and we have Si(G1)=Si(G2)(i=0,1,…,k-1) and Sk(G1)<Sk(G2).In this paper,we determine the second to the fourth largest quasi-unicyclic graphs,in an S-order,in the set Q(n,d0),respectively.
