The exponential Randić index has important applications in the fields of biology and chemistry. The exponential Randić index of a graph G is defined as the sum of the weights e 1 d( u )d( v ) of all edges uv of G, whe...The exponential Randić index has important applications in the fields of biology and chemistry. The exponential Randić index of a graph G is defined as the sum of the weights e 1 d( u )d( v ) of all edges uv of G, where d( u ) denotes the degree of a vertex u in G. The paper mainly provides the upper and lower bounds of the exponential Randić index in quasi-tree graphs, and characterizes the extremal graphs when the bounds are achieved.展开更多
A connected graph G=(V,E)is called a quasi-tree graph if there exists a vertex v0∈V(G)such that G-v0 is a tree.In this paper,we determine all quasi-tree graphs of order n with the second largest signless Laplacian ei...A connected graph G=(V,E)is called a quasi-tree graph if there exists a vertex v0∈V(G)such that G-v0 is a tree.In this paper,we determine all quasi-tree graphs of order n with the second largest signless Laplacian eigenvalue greater than or equal to n-3.As an application,we determine all quasi-tree graphs of order n with the sum of the two largest signless Laplacian eigenvalues greater than to 2 n-5/4.展开更多
A connected graph G = （V, E） is called a quasi-tree graph, if there exists a vertex vo ∈ V（G） such that G - v0 is a tree. Liu and Lu [Linear Algebra Appl. 428 （2008） 2708- 2714] determined the maximal spectral ...A connected graph G = （V, E） is called a quasi-tree graph, if there exists a vertex vo ∈ V（G） such that G - v0 is a tree. Liu and Lu [Linear Algebra Appl. 428 （2008） 2708- 2714] determined the maximal spectral radius together with the corresponding graph among all quasi-tree graphs on n vertices. In this paper, we extend their result, and determine the second to the fifth largest spectral radii together with the corresponding graphs among all quasi-tree graphs on n vertices.展开更多
A maximal independent set is an independent set that is not a proper subset of any other independent set. A connected graph (respectively, graph) G with vertex set V(G) is called a quasi-tree graph (respectively, quas...A maximal independent set is an independent set that is not a proper subset of any other independent set. A connected graph (respectively, graph) G with vertex set V(G) is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex x ∈V(G) such that G −x?is a tree (respectively, forest). In this paper, we survey on the large numbers of maximal independent sets among all trees, forests, quasi-trees and quasi-forests. In addition, we further look into the problem of determining the third largest number of maximal independent sets among all quasi-trees and quasi-forests. Extremal graphs achieving these values are also given.展开更多
For a graph G,the first leap Zagreb index is defined as LM_(1)(G)=∑_(ν∈V(G))d_(2)(v/G)^(2),where d_(2)(v/G) is the 2-distance degree of a vertex v in G.Let QT^(k)(n) be the set of kgeneralized quasi-trees with n ve...For a graph G,the first leap Zagreb index is defined as LM_(1)(G)=∑_(ν∈V(G))d_(2)(v/G)^(2),where d_(2)(v/G) is the 2-distance degree of a vertex v in G.Let QT^(k)(n) be the set of kgeneralized quasi-trees with n vertices.In this paper,we determine the extremal elements from the set QT^(k)(n) with respect to the first leap Zagreb index.展开更多
文摘The exponential Randić index has important applications in the fields of biology and chemistry. The exponential Randić index of a graph G is defined as the sum of the weights e 1 d( u )d( v ) of all edges uv of G, where d( u ) denotes the degree of a vertex u in G. The paper mainly provides the upper and lower bounds of the exponential Randić index in quasi-tree graphs, and characterizes the extremal graphs when the bounds are achieved.
基金Supported by the National Natural Science Foundation of China(Grant No.11771443)the Fundamental Research Funds for the Central Universities(Grant No.2018BSCXB24)the Postgraduate Research&Practice Innovation Program of Jiangsu Province(Grant No.KYCX18JL980).
文摘A connected graph G=(V,E)is called a quasi-tree graph if there exists a vertex v0∈V(G)such that G-v0 is a tree.In this paper,we determine all quasi-tree graphs of order n with the second largest signless Laplacian eigenvalue greater than or equal to n-3.As an application,we determine all quasi-tree graphs of order n with the sum of the two largest signless Laplacian eigenvalues greater than to 2 n-5/4.
基金Supported by the National Natural Science Foundation of China(Grant No.11171290)the Natural Science Foundation of Jiangsu Province(Grant No.BK20151295)
文摘A connected graph G = （V, E） is called a quasi-tree graph, if there exists a vertex vo ∈ V（G） such that G - v0 is a tree. Liu and Lu [Linear Algebra Appl. 428 （2008） 2708- 2714] determined the maximal spectral radius together with the corresponding graph among all quasi-tree graphs on n vertices. In this paper, we extend their result, and determine the second to the fifth largest spectral radii together with the corresponding graphs among all quasi-tree graphs on n vertices.
文摘A maximal independent set is an independent set that is not a proper subset of any other independent set. A connected graph (respectively, graph) G with vertex set V(G) is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex x ∈V(G) such that G −x?is a tree (respectively, forest). In this paper, we survey on the large numbers of maximal independent sets among all trees, forests, quasi-trees and quasi-forests. In addition, we further look into the problem of determining the third largest number of maximal independent sets among all quasi-trees and quasi-forests. Extremal graphs achieving these values are also given.
基金Supported by the Foundation of Henan Department of Science and Technology(Grant No.182102310830)the Foundation of Henan University of Engineering(Grant No.D2016018)+1 种基金the Foundation of Henan Educational Committee(Grant Nos.20A1100162020GGJS239)。
文摘For a graph G,the first leap Zagreb index is defined as LM_(1)(G)=∑_(ν∈V(G))d_(2)(v/G)^(2),where d_(2)(v/G) is the 2-distance degree of a vertex v in G.Let QT^(k)(n) be the set of kgeneralized quasi-trees with n vertices.In this paper,we determine the extremal elements from the set QT^(k)(n) with respect to the first leap Zagreb index.
