The harmonic index of a graph?G? is defined as where d(u) denotes the degree of a vertex u in G . In this work, we give another expression for the Harmonic index. Using this expression, we give the minimum value of th...The harmonic index of a graph?G? is defined as where d(u) denotes the degree of a vertex u in G . In this work, we give another expression for the Harmonic index. Using this expression, we give the minimum value of the harmonic index for any triangle-free graphs with order n and minimum degree δ ≥ k for k≤ n/2? and show the corresponding extremal graph is the complete graph.展开更多
A graph G is called triangle-free if G does not contain any triangle as its induced subgraph.Let G_(n)be the set of triangle-free graphs of order n each of which has three positive eigenvalues.In this paper,we find 20...A graph G is called triangle-free if G does not contain any triangle as its induced subgraph.Let G_(n)be the set of triangle-free graphs of order n each of which has three positive eigenvalues.In this paper,we find 20 specific graphs in G_(n),each of which has nullity no more than 2,and we show that in terms of three graph transformations all the other graphs of G_(n)can be constructed from these 20 specific graphs.Hence,we completely characterize the triangle-free graphs with exactly three positive eigenvalues.展开更多
Let G be a connected graph with order n,minimum degree δ = δ（G） and edge-connectivity λ =λ（G）. A graph G is maximally edge-connected if λ = δ, and super edge-connected if every minimum edgecut consists of ed...Let G be a connected graph with order n,minimum degree δ = δ（G） and edge-connectivity λ =λ（G）. A graph G is maximally edge-connected if λ = δ, and super edge-connected if every minimum edgecut consists of edges incident with a vertex of minimum degree. Define the zeroth-order general Randic index R_α-0（G） =Σ x∈V（G） d_G-α（x）, where dG（x） denotes the degree of the vertex x. In this paper, we present two sufficient conditions for graphs and triangle-free graphs to be super edge-connected in terms of the zeroth-order general Randic index for -1 ≤α 〈 0, respectively.展开更多
A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v, denoted by dw（v...A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v, denoted by dw（v）. The weight of a cycle is defined as the sum of the weights of its edges. Fujisawa proved that if G is a 2-connected triangle-free weighted graph such that the minimum weighted degree of G is at least d, then G contains a cycle of weight at least 2d. In this paper, we proved that if G is a 2-connected triangle-free weighted graph of even size such that dw（u） ＋ dw（v） ≥ 2d holds for any pair of nonadjacent vertices u, v ∈ V（G）, then G contains a cycle of weight at least 2d.展开更多
设G=(V,E)是一个连通图,边集S(?)E是一个3-限制性边割,如果G-S是不连通的并且G-S的每个分支至少有三个点.图G的3-限制性边连通度λ_3(G)是G中最小的一个3-限制性边割的基数.图G是λ_3(G)连通的,如果3-限制性边割存在.G是λ_3-最优的,如...设G=(V,E)是一个连通图,边集S(?)E是一个3-限制性边割,如果G-S是不连通的并且G-S的每个分支至少有三个点.图G的3-限制性边连通度λ_3(G)是G中最小的一个3-限制性边割的基数.图G是λ_3(G)连通的,如果3-限制性边割存在.G是λ_3-最优的,如果λ_3(G)=ξ_3(G),其中ξ_3(G)=min{|[U,(?)]|:U(?)V,|U|=3 and G[U]是连通的).G[U]表示V的子集U的导出子图,(?)=V\U表示U的补.[U,(?)]是一条边的一个端点在U中另一个端点在(?)中的边的集合.本文给出了不含三角形的图是λ_3-最优的一些充分条件.展开更多
Define the zeroth-order Randic index R^(0)(G)=∑x∈V(G)1/√dG1(x),where dG(x)denotes the degree of the vertex x.In this paper,we present two sufficient conditions for graphs and triangle-free graphs to be super-edge-c...Define the zeroth-order Randic index R^(0)(G)=∑x∈V(G)1/√dG1(x),where dG(x)denotes the degree of the vertex x.In this paper,we present two sufficient conditions for graphs and triangle-free graphs to be super-edge-connected in terms of the zeroth-order Randic index,respectively.展开更多
文摘The harmonic index of a graph?G? is defined as where d(u) denotes the degree of a vertex u in G . In this work, we give another expression for the Harmonic index. Using this expression, we give the minimum value of the harmonic index for any triangle-free graphs with order n and minimum degree δ ≥ k for k≤ n/2? and show the corresponding extremal graph is the complete graph.
基金Supported by the Natural Science Foundation of Xinjiang Uygur Autonomous Region(No.2022D01A218)the Doctoral Scientific Research Foundation of Xinjiang Normal University(No.XJNUBS2009).
文摘A graph G is called triangle-free if G does not contain any triangle as its induced subgraph.Let G_(n)be the set of triangle-free graphs of order n each of which has three positive eigenvalues.In this paper,we find 20 specific graphs in G_(n),each of which has nullity no more than 2,and we show that in terms of three graph transformations all the other graphs of G_(n)can be constructed from these 20 specific graphs.Hence,we completely characterize the triangle-free graphs with exactly three positive eigenvalues.
基金supported by the National Natural Science Foundation of China(No.11501490,61373019,11371307)by the Natural Science Foundation of Shandong Province(No.ZR2015AM006)
文摘Let G be a connected graph with order n,minimum degree δ = δ（G） and edge-connectivity λ =λ（G）. A graph G is maximally edge-connected if λ = δ, and super edge-connected if every minimum edgecut consists of edges incident with a vertex of minimum degree. Define the zeroth-order general Randic index R_α-0（G） =Σ x∈V（G） d_G-α（x）, where dG（x） denotes the degree of the vertex x. In this paper, we present two sufficient conditions for graphs and triangle-free graphs to be super edge-connected in terms of the zeroth-order general Randic index for -1 ≤α 〈 0, respectively.
基金Supported by National Natural Science Foundation of China(Grant No.11001269)
文摘A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v, denoted by dw（v）. The weight of a cycle is defined as the sum of the weights of its edges. Fujisawa proved that if G is a 2-connected triangle-free weighted graph such that the minimum weighted degree of G is at least d, then G contains a cycle of weight at least 2d. In this paper, we proved that if G is a 2-connected triangle-free weighted graph of even size such that dw（u） ＋ dw（v） ≥ 2d holds for any pair of nonadjacent vertices u, v ∈ V（G）, then G contains a cycle of weight at least 2d.
基金supported by NSFC (No.10671165)XJEDU (No.2004G05).
文摘设G=(V,E)是一个连通图,边集S(?)E是一个3-限制性边割,如果G-S是不连通的并且G-S的每个分支至少有三个点.图G的3-限制性边连通度λ_3(G)是G中最小的一个3-限制性边割的基数.图G是λ_3(G)连通的,如果3-限制性边割存在.G是λ_3-最优的,如果λ_3(G)=ξ_3(G),其中ξ_3(G)=min{|[U,(?)]|:U(?)V,|U|=3 and G[U]是连通的).G[U]表示V的子集U的导出子图,(?)=V\U表示U的补.[U,(?)]是一条边的一个端点在U中另一个端点在(?)中的边的集合.本文给出了不含三角形的图是λ_3-最优的一些充分条件.
基金Supported by the Natural Science Special Research Foundation of the Education Department of Shaanxi Province(No.12JK089)the Science Research Foundation of Shangluo University(No.12SKY011)
基金This work is supported by the National Natural Science Foundation of China(Nos.11501490,61373019,13071107)the Natural Science Foundation of Shandong Province(No.ZR2015AM006).
文摘Define the zeroth-order Randic index R^(0)(G)=∑x∈V(G)1/√dG1(x),where dG(x)denotes the degree of the vertex x.In this paper,we present two sufficient conditions for graphs and triangle-free graphs to be super-edge-connected in terms of the zeroth-order Randic index,respectively.
