The atom-bond connectivity（ABC） index provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes,which is defined as ABC（G） =∑ uv∈E（G） √d u＋dv-2 dudv,...The atom-bond connectivity（ABC） index provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes,which is defined as ABC（G） =∑ uv∈E（G） √d u＋dv-2 dudv,where du denotes the degree of a vertex u in G.A chemical graph is a graph in which no vertex has degree greater than 4.In this paper,we obtain the sharp upper and lower bounds on ABC index of chemical bicyclic graphs.展开更多
A graph G of order n is called a bicyclic graph if G is connected and the number of edges of G is n＋1. Let B（n） be the set of all bicyclic graphs on n vertices. In this paper, the first three largest spectral radii...A graph G of order n is called a bicyclic graph if G is connected and the number of edges of G is n＋1. Let B（n） be the set of all bicyclic graphs on n vertices. In this paper, the first three largest spectral radii in the class B（n） （n ≥9） together with the corresponding graphs are given.展开更多
The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In our recent work, we have determined the graphs...The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In our recent work, we have determined the graphs with maximal Laplacian spreads among all trees of fixed order and among all unicyclic graphs of fixed order, respectively. In this paper, we continue the work on Laplacian spread of graphs, and prove that there exist exactly two bicyclic graphs with maximal Laplacian spread among all bicyclic graphs of fixed order, which are obtained from a star by adding two incident edges and by adding two nonincident edges between the pendant vertices of the star, respectively.展开更多
A graph has exactly two main eigenvalues if and only if it is a 2-walk linear graph.In this paper,we show some necessary conditions that a 2-walk（a,b）-linear graph must obey.Using these conditions and some basic the...A graph has exactly two main eigenvalues if and only if it is a 2-walk linear graph.In this paper,we show some necessary conditions that a 2-walk（a,b）-linear graph must obey.Using these conditions and some basic theorems in graph theory,we characterize all 2-walk linear graphs with small cyclic graphs without pendants.The results are given in sort on unicyclic,bicyclic,tricyclic graphs.展开更多
The spectral spread of a graph is defined to be the difference between the largest and the least eigenvalue of the adjacency matrix of the graph. A graph G is said to be bicyclic, if G is connected and ｜E（G）｜ = ...The spectral spread of a graph is defined to be the difference between the largest and the least eigenvalue of the adjacency matrix of the graph. A graph G is said to be bicyclic, if G is connected and ｜E（G）｜ = ｜V （G）｜ ＋ 1. Let B（n, g） be the set of bicyclic graphs on n vertices with girth g. In this paper some properties about the least eigenvalues of graphs are given, by which the unique graph with maximal spectral spread in B（n, g） is determined.展开更多
A graph G is nonsingular if its adjacency matrix A(G)is nonsingular.A nonsingular graph G is said to have an inverse G+if A(G)-1 is signature similar to a nonnegative matrix.Let H be the class of connected bipartite g...A graph G is nonsingular if its adjacency matrix A(G)is nonsingular.A nonsingular graph G is said to have an inverse G+if A(G)-1 is signature similar to a nonnegative matrix.Let H be the class of connected bipartite graphs with unique perfect matchings.We present a characterization of bicyclic graphs in H which possess unicyclic or bicyclic inverses.展开更多
基金Supported by the National Natural Science Foundation of China(11071272,10831001,11171279,11101087)the Young Talent Foundation of Fuzhou University(XRC-1154)
文摘The atom-bond connectivity（ABC） index provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes,which is defined as ABC（G） =∑ uv∈E（G） √d u＋dv-2 dudv,where du denotes the degree of a vertex u in G.A chemical graph is a graph in which no vertex has degree greater than 4.In this paper,we obtain the sharp upper and lower bounds on ABC index of chemical bicyclic graphs.
基金the National Natural Science Foundation of China(10331020).
文摘A graph G of order n is called a bicyclic graph if G is connected and the number of edges of G is n＋1. Let B（n） be the set of all bicyclic graphs on n vertices. In this paper, the first three largest spectral radii in the class B（n） （n ≥9） together with the corresponding graphs are given.
基金Supported by the National Natural Science Foundation of China (Grant No.10601001)the Natural Science Foundation of Anhui Province (Grant No.070412065)+3 种基金Project of Anhui Province for Young Teachers Research Supportin Universities (Grant No.2008jql083)Natural Science Foundation of Department of Education of Anhui Province(Grant No.2005kj005zd)Project of Anhui University on Leading Researchers ConstructionFoundation of Innovation Team on Basic Mathematics of Anhui University
文摘The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In our recent work, we have determined the graphs with maximal Laplacian spreads among all trees of fixed order and among all unicyclic graphs of fixed order, respectively. In this paper, we continue the work on Laplacian spread of graphs, and prove that there exist exactly two bicyclic graphs with maximal Laplacian spread among all bicyclic graphs of fixed order, which are obtained from a star by adding two incident edges and by adding two nonincident edges between the pendant vertices of the star, respectively.
基金Supported by the National Natural Science Foundation of China (10671081)
文摘A graph has exactly two main eigenvalues if and only if it is a 2-walk linear graph.In this paper,we show some necessary conditions that a 2-walk（a,b）-linear graph must obey.Using these conditions and some basic theorems in graph theory,we characterize all 2-walk linear graphs with small cyclic graphs without pendants.The results are given in sort on unicyclic,bicyclic,tricyclic graphs.
基金Supported by the National Natural Science Foundation of China(No.11101057)China Postdoctoral Science Foundation(No.20110491443)+1 种基金the NSF of Education Ministry of Anhui province(No.KJ2012Z283)Scientific Research Foundation of Chuzhou University(No.2011kj004B)
文摘The spectral spread of a graph is defined to be the difference between the largest and the least eigenvalue of the adjacency matrix of the graph. A graph G is said to be bicyclic, if G is connected and ｜E（G）｜ = ｜V （G）｜ ＋ 1. Let B（n, g） be the set of bicyclic graphs on n vertices with girth g. In this paper some properties about the least eigenvalues of graphs are given, by which the unique graph with maximal spectral spread in B（n, g） is determined.
基金NSFC(Grant No.11761070,61662079)Postgraduate Innovation Project of Xinjiang,Xinjiang Normal University Undergraduate teaching project(SDJG2017-3)The Opening Project of Key Laboratory of Xinjiang Normal University(No:XJNUSYS082017A02).
文摘A graph G is nonsingular if its adjacency matrix A(G)is nonsingular.A nonsingular graph G is said to have an inverse G+if A(G)-1 is signature similar to a nonnegative matrix.Let H be the class of connected bipartite graphs with unique perfect matchings.We present a characterization of bicyclic graphs in H which possess unicyclic or bicyclic inverses.
