Let be a graph with n vertices and m edges. The sum of absolute value of all coefficients of matching polynomial is called Hosoya index. In this paper, we determine 2<sup>nd</sup> to 4<sup>th</sup...Let be a graph with n vertices and m edges. The sum of absolute value of all coefficients of matching polynomial is called Hosoya index. In this paper, we determine 2<sup>nd</sup> to 4<sup>th</sup> minimum Hosoya index of a kind of tetracyclic graph, with m = n +3.展开更多
Let G be a （molecular） graph. The Hosoya index Z（G） of G is defined as the number of subsets of the edge set E（G） in which no two edges are adjacent in G, i.e., Z（G） is the total number of matchings of G. In t...Let G be a （molecular） graph. The Hosoya index Z（G） of G is defined as the number of subsets of the edge set E（G） in which no two edges are adjacent in G, i.e., Z（G） is the total number of matchings of G. In this paper, we determine all the connected graphs G with n ＋ 1 ≤ Z（G） ≤5n - 17 for n ≥ 19. As a byproduct, the graphs of n vertices with Hosoya index from the second smallest value to the twenty first smallest value are obtained for n ≥ 19.展开更多
Let G be a simple graph. Define R(G) to be the graph obtained from G by adding a new vertex e* corresponding to each edge e = (a, b) of G and by joining each new vertex e* to the end vertices a and b of the edge e cor...Let G be a simple graph. Define R(G) to be the graph obtained from G by adding a new vertex e* corresponding to each edge e = (a, b) of G and by joining each new vertex e* to the end vertices a and b of the edge e corresponding to it. In this paper, we prove that the number of matchings of R(G) is completely determined by the degree sequence of vertices of G.展开更多
The Hosoya index of a graph is defined as the total number of the matching of the graph. In this paper, the ordering of polygonal chains with respect to Hosoya index is characterized.
The Hosoya index of a graph is the total number of matchings in it. And the Merrifield-Simmons index is the total number of independent sets in it. They are typical examples of graph invariants used in mathematical ch...The Hosoya index of a graph is the total number of matchings in it. And the Merrifield-Simmons index is the total number of independent sets in it. They are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. In this paper, we obtain explicit analytical expressions for the expectations of the Hosoya index and the Merrifield-Simmons index of a random polyphenyl chain.展开更多
The Merrifield-Simmons index of a graph is defined as the total number of the independent sets of the graph and the Ho- soya index of a graph is defined as the total number of the match- ings of the graph. In this pap...The Merrifield-Simmons index of a graph is defined as the total number of the independent sets of the graph and the Ho- soya index of a graph is defined as the total number of the match- ings of the graph. In this paper, the definition of a class of po- lygonal chains is given, ordering of the polygonal chains with respect to Merrifield-Simmons index and Hosoya index are ob- tained, and their extremal graphs with respect to these two topo- logical indices are determined.展开更多
Let n and d be two positive integers.By B_(n,d) we denote the graph obtained by identifying an endvertex of path P_d with the center of star S_(n-d+1),where n ≥ d + 1.By C_(n,d) we denote the graph obtained by identi...Let n and d be two positive integers.By B_(n,d) we denote the graph obtained by identifying an endvertex of path P_d with the center of star S_(n-d+1),where n ≥ d + 1.By C_(n,d) we denote the graph obtained by identifying an endvertex of P_(d-1) with the center of Stare S_(n-d),and the other endvertex of P_(d-1) with the center of S_3 where n ≥ d + 3.By E_(n,d,k) we denote the graph obtained by identifying the vertex v_k of P(v_1 - v_2 - ··· - v_(d+1)) with the center of S_(n-d).In this paper,we completely characterize all trees T which have diameter at least d(d ≥ 3) and satisfy the following conditions:(i) Z(B_(n,d)) ≤ Z(T) ≤ Z(E_(n,d,3)) for n = d + 3;(ii) Z(B_(n,d)) ≤ Z(T) ≤ Z(C_(n,d)) for n ≥ d + 4.展开更多
Chemical compounds are modeled as graphs.The atoms of molecules represent the graph vertices while chemical bonds between the atoms express the edges.The topological indices representing the molecular graph correspond...Chemical compounds are modeled as graphs.The atoms of molecules represent the graph vertices while chemical bonds between the atoms express the edges.The topological indices representing the molecular graph corresponds to the different chemical properties of compounds.Let a,b be are two positive integers,andΓ(Z_(a)×Z_(b))be the zero-divisor graph of the commutative ring Z_(a)×Z_(b).In this article some direct questions have been answered that can be utilized latterly in different applications.This study starts with simple computations,leading to a quite complex ring theoretic problems to prove certain properties.The theory of finite commutative rings is useful due to its different applications in the fields of advanced mechanics,communication theory,cryptography,combinatorics,algorithms analysis,and engineering.In this paper we determine the distance-based topological polynomials and indices of the zero-divisor graph of the commutative ring Z_(p^(2))×Z_(q)(for p,q as prime numbers)with the help of graphical structure analysis.The study outcomes help in understanding the fundamental relation between ring-theoretic and graph-theoretic properties of a zero-divisor graphΓ(G).展开更多
The Merrifield-Simmons index and Hosoya index are defined as the number of the graph G（V, E） as the number of subsets of V（G） in which no tow vertices are adjacent and the number of subsets of E（G） in which no t...The Merrifield-Simmons index and Hosoya index are defined as the number of the graph G（V, E） as the number of subsets of V（G） in which no tow vertices are adjacent and the number of subsets of E（G） in which no two edges are incident, respectively. In this paper, we characterize the Unicyclic graphs with Merrifield-Simmons indices and Hosoya indices, respectively. And double-cyclic graphs with Hosoya indices among the doublecyclic graphs with n vertices.展开更多
The Hosoya index of a graph is the total number of matchings in it.And the Merrifield-Simmons index is the total number of independent sets in it.They are typical examples of graph invariants used in mathematical chem...The Hosoya index of a graph is the total number of matchings in it.And the Merrifield-Simmons index is the total number of independent sets in it.They are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure.In this paper,we obtain explicit analytical expressions for the expectations of the Hosoya index and the Merrifield-Simmons index of a random polyphenyl chain.展开更多
假设图 G 是一个简单无向图,则图 G的 Hosoya 指标指的是该图的所有匹配数目之和.令I是单位矩阵,A(G) 是图 G 的邻接矩阵,则积和多项式定义为 π(G,x)= per(xI-A(G)).图 G 的永久和定义为其积和多项式系数的绝对值之和.本文研究了三圈...假设图 G 是一个简单无向图,则图 G的 Hosoya 指标指的是该图的所有匹配数目之和.令I是单位矩阵,A(G) 是图 G 的邻接矩阵,则积和多项式定义为 π(G,x)= per(xI-A(G)).图 G 的永久和定义为其积和多项式系数的绝对值之和.本文研究了三圈图中玫瑰图的 Hosoya 的第二小并刻画了极图:其次,研究了玫瑰图永久和的第二小并刻画了极图。Let G be a simple and undirected graph. The Hosoya index Z(G) of G is defined to be the total number of matchings of G. Let I be an identity matrix and A(G) be an adjacency matrix of G. Then the permanental polynomial of G is defined asπ(G,x) = per(xI-A(G)). The permanental sum of G is defined as the sum of the absolutevalues of all coefficients of π(G,x). In this paper, we prove the second smallest Hosoyaindex of rose graphs and determine the extremal graphs. Besides, we prove the secondsmallest permanental sum of rose graphs and determine the extremal graphs.展开更多
文摘Let be a graph with n vertices and m edges. The sum of absolute value of all coefficients of matching polynomial is called Hosoya index. In this paper, we determine 2<sup>nd</sup> to 4<sup>th</sup> minimum Hosoya index of a kind of tetracyclic graph, with m = n +3.
基金Supported by the National Natural Science Foundation of China(10761008, 10461009)the Science Foundation of the State Education Ministry of China(205170)
文摘Let G be a （molecular） graph. The Hosoya index Z（G） of G is defined as the number of subsets of the edge set E（G） in which no two edges are adjacent in G, i.e., Z（G） is the total number of matchings of G. In this paper, we determine all the connected graphs G with n ＋ 1 ≤ Z（G） ≤5n - 17 for n ≥ 19. As a byproduct, the graphs of n vertices with Hosoya index from the second smallest value to the twenty first smallest value are obtained for n ≥ 19.
文摘Let G be a simple graph. Define R(G) to be the graph obtained from G by adding a new vertex e* corresponding to each edge e = (a, b) of G and by joining each new vertex e* to the end vertices a and b of the edge e corresponding to it. In this paper, we prove that the number of matchings of R(G) is completely determined by the degree sequence of vertices of G.
基金Supported by the National Natural Science Foundation of China(10761008)the Scientific Research Foundation of the Education Department of Guangxi Province of China(201010LX471,201010LX495,201106LX595,201106LX608)the Natural Science Fund of Hechi University(2011YBZ-N003,2012YBZ-N004)
文摘The Hosoya index of a graph is defined as the total number of the matching of the graph. In this paper, the ordering of polygonal chains with respect to Hosoya index is characterized.
基金Supported by the Fundamental Research Funds for the Central Universities(No.20720160038)Research Projects for Young and Middle-aged Teachers of Fujian Province(No.JA15016)
文摘The Hosoya index of a graph is the total number of matchings in it. And the Merrifield-Simmons index is the total number of independent sets in it. They are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. In this paper, we obtain explicit analytical expressions for the expectations of the Hosoya index and the Merrifield-Simmons index of a random polyphenyl chain.
基金Supported by the National Natural Science Foundation of China(11161041)Innovative Team Subsidize of Northwest University for Nationalitiesthe Fundamental Research Funds for the Central Universities(31920140059)
文摘The Merrifield-Simmons index of a graph is defined as the total number of the independent sets of the graph and the Ho- soya index of a graph is defined as the total number of the match- ings of the graph. In this paper, the definition of a class of po- lygonal chains is given, ordering of the polygonal chains with respect to Merrifield-Simmons index and Hosoya index are ob- tained, and their extremal graphs with respect to these two topo- logical indices are determined.
基金Foundation item： Supported by the National Science Foundation of China（11161037） Supported by the National Science Foundation of Qinghai（2011-z-907）
文摘Let n and d be two positive integers.By B_(n,d) we denote the graph obtained by identifying an endvertex of path P_d with the center of star S_(n-d+1),where n ≥ d + 1.By C_(n,d) we denote the graph obtained by identifying an endvertex of P_(d-1) with the center of Stare S_(n-d),and the other endvertex of P_(d-1) with the center of S_3 where n ≥ d + 3.By E_(n,d,k) we denote the graph obtained by identifying the vertex v_k of P(v_1 - v_2 - ··· - v_(d+1)) with the center of S_(n-d).In this paper,we completely characterize all trees T which have diameter at least d(d ≥ 3) and satisfy the following conditions:(i) Z(B_(n,d)) ≤ Z(T) ≤ Z(E_(n,d,3)) for n = d + 3;(ii) Z(B_(n,d)) ≤ Z(T) ≤ Z(C_(n,d)) for n ≥ d + 4.
文摘Chemical compounds are modeled as graphs.The atoms of molecules represent the graph vertices while chemical bonds between the atoms express the edges.The topological indices representing the molecular graph corresponds to the different chemical properties of compounds.Let a,b be are two positive integers,andΓ(Z_(a)×Z_(b))be the zero-divisor graph of the commutative ring Z_(a)×Z_(b).In this article some direct questions have been answered that can be utilized latterly in different applications.This study starts with simple computations,leading to a quite complex ring theoretic problems to prove certain properties.The theory of finite commutative rings is useful due to its different applications in the fields of advanced mechanics,communication theory,cryptography,combinatorics,algorithms analysis,and engineering.In this paper we determine the distance-based topological polynomials and indices of the zero-divisor graph of the commutative ring Z_(p^(2))×Z_(q)(for p,q as prime numbers)with the help of graphical structure analysis.The study outcomes help in understanding the fundamental relation between ring-theoretic and graph-theoretic properties of a zero-divisor graphΓ(G).
基金This project is supported by National Natural Science Foundation of China（10671081） and the Science Foundation of Hubei Province（2006AA412C27）
文摘The Merrifield-Simmons index and Hosoya index are defined as the number of the graph G（V, E） as the number of subsets of V（G） in which no tow vertices are adjacent and the number of subsets of E（G） in which no two edges are incident, respectively. In this paper, we characterize the Unicyclic graphs with Merrifield-Simmons indices and Hosoya indices, respectively. And double-cyclic graphs with Hosoya indices among the doublecyclic graphs with n vertices.
基金by the Fundamental Research Funds for the Central Universities(Grant No.20720190071).
文摘The Hosoya index of a graph is the total number of matchings in it.And the Merrifield-Simmons index is the total number of independent sets in it.They are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure.In this paper,we obtain explicit analytical expressions for the expectations of the Hosoya index and the Merrifield-Simmons index of a random polyphenyl chain.
文摘假设图 G 是一个简单无向图,则图 G的 Hosoya 指标指的是该图的所有匹配数目之和.令I是单位矩阵,A(G) 是图 G 的邻接矩阵,则积和多项式定义为 π(G,x)= per(xI-A(G)).图 G 的永久和定义为其积和多项式系数的绝对值之和.本文研究了三圈图中玫瑰图的 Hosoya 的第二小并刻画了极图:其次,研究了玫瑰图永久和的第二小并刻画了极图。Let G be a simple and undirected graph. The Hosoya index Z(G) of G is defined to be the total number of matchings of G. Let I be an identity matrix and A(G) be an adjacency matrix of G. Then the permanental polynomial of G is defined asπ(G,x) = per(xI-A(G)). The permanental sum of G is defined as the sum of the absolutevalues of all coefficients of π(G,x). In this paper, we prove the second smallest Hosoyaindex of rose graphs and determine the extremal graphs. Besides, we prove the secondsmallest permanental sum of rose graphs and determine the extremal graphs.
