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
Given a connected graph G,the revised edge-revised Szeged index is defined as Sz_(e)^(*)(G)=∑_(e=uv∈E_(G))(m_(u)(e)+m_(0)(e)/2)(m_(v)(e)+m_(0)(e)/w),where m_(u)(e),m_(v)(e)and m_(0)(e)are the number of edges of G ly...Given a connected graph G,the revised edge-revised Szeged index is defined as Sz_(e)^(*)(G)=∑_(e=uv∈E_(G))(m_(u)(e)+m_(0)(e)/2)(m_(v)(e)+m_(0)(e)/w),where m_(u)(e),m_(v)(e)and m_(0)(e)are the number of edges of G lying closer to vertex u than to vertex u,the number of edges of G lying closer to vertex than to vertex u and the number of edges of G at the same distance to u and u,respectively.In this paper,by transformation and calculation,the lower bound of revised edge-Szeged index of unicyclic graphs with given diameter is obtained,and the extremal graph is depicted.展开更多
An H-free graph is a graph not containing the given graph H as a subgraph.It is well known that the Turán number ex(n,H)is the maximum number of edges in an H-free graph on n vertices.Based on this definition,we ...An H-free graph is a graph not containing the given graph H as a subgraph.It is well known that the Turán number ex(n,H)is the maximum number of edges in an H-free graph on n vertices.Based on this definition,we define ex_(P)(n,H)to restrict the graph classes to planar graphs,that is,ex_(P)(n,H)=max{|E(G)|:G∈G,where G is a family of all H-free planar graphs on n vertices.Obviously,we have ex_(P)(n,H)=3n−6 if the graph H is not a planar graph.The study is initiated by Dowden(J Graph Theory 83:213–230,2016),who obtained some results when H is considered as C_(4)or C_(5).In this paper,we determine the values of ex_(P)(n,Pk)with k∈{8,9},where Pk is a path with k 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
Let F={H_(1),...,H_(k)}(k≥1)be a family of graphs.The Tur´an number of the family F is the maximum number of edges in an n-vertex{H_(1),...,H_(k)}-free graph,denoted by ex(n,F)or ex(n,{H_(1),H_(2),...,H_(k)}).Th...Let F={H_(1),...,H_(k)}(k≥1)be a family of graphs.The Tur´an number of the family F is the maximum number of edges in an n-vertex{H_(1),...,H_(k)}-free graph,denoted by ex(n,F)or ex(n,{H_(1),H_(2),...,H_(k)}).The blow-up of a graph H is the graph obtained from H by replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different.In this paper we determine the Tur´an number of the family consisting of a blow-up of a cycle and a blow-up of a star in terms of the Tur´an number of the family consisting of a cycle,a star and linear forests with k edges.展开更多
The problem studied in this paper is to determine E(p,C),the maximum size of a connected graph G with the given vertex number p and cutwidth C. This paper presents some results on this problem.
The circular chromatic number and the fractional chromatic number are two generalizations of the ordinary chromatic number of a graph. A graph is called star extremal if its fractional chromatic number equals to its c...The circular chromatic number and the fractional chromatic number are two generalizations of the ordinary chromatic number of a graph. A graph is called star extremal if its fractional chromatic number equals to its circular chromatic number (also known as the star chromatic number). This paper studies the star extremality of the circulant graphs whose generating sets are of the form {±1,±k} . 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.
文摘Given a connected graph G,the revised edge-revised Szeged index is defined as Sz_(e)^(*)(G)=∑_(e=uv∈E_(G))(m_(u)(e)+m_(0)(e)/2)(m_(v)(e)+m_(0)(e)/w),where m_(u)(e),m_(v)(e)and m_(0)(e)are the number of edges of G lying closer to vertex u than to vertex u,the number of edges of G lying closer to vertex than to vertex u and the number of edges of G at the same distance to u and u,respectively.In this paper,by transformation and calculation,the lower bound of revised edge-Szeged index of unicyclic graphs with given diameter is obtained,and the extremal graph is depicted.
基金the National Natural Science Foundation of China(Nos.11922112 and 11771221)the Natural Science Foundation of Tianjin(Nos.20JCZDJC00840 and 20JCJQJC00090)+2 种基金Yong-Xin Lan was partially supported by the National Natural Science Foundation of China(No.12001154)the Natural Science Foundation of Hebei Province(No.A2021202025)the Special Funds for Jointly Building Universities of Tianjin(No.280000307).
文摘An H-free graph is a graph not containing the given graph H as a subgraph.It is well known that the Turán number ex(n,H)is the maximum number of edges in an H-free graph on n vertices.Based on this definition,we define ex_(P)(n,H)to restrict the graph classes to planar graphs,that is,ex_(P)(n,H)=max{|E(G)|:G∈G,where G is a family of all H-free planar graphs on n vertices.Obviously,we have ex_(P)(n,H)=3n−6 if the graph H is not a planar graph.The study is initiated by Dowden(J Graph Theory 83:213–230,2016),who obtained some results when H is considered as C_(4)or C_(5).In this paper,we determine the values of ex_(P)(n,Pk)with k∈{8,9},where Pk is a path with k 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 National Nature Science Foundation of China(Grant Nos.11871329,11971298)。
文摘Let F={H_(1),...,H_(k)}(k≥1)be a family of graphs.The Tur´an number of the family F is the maximum number of edges in an n-vertex{H_(1),...,H_(k)}-free graph,denoted by ex(n,F)or ex(n,{H_(1),H_(2),...,H_(k)}).The blow-up of a graph H is the graph obtained from H by replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different.In this paper we determine the Tur´an number of the family consisting of a blow-up of a cycle and a blow-up of a star in terms of the Tur´an number of the family consisting of a cycle,a star and linear forests with k edges.
基金Supported by Natural Science Foundation of Zhejiang Province(1 0 2 0 5 5 )
文摘The problem studied in this paper is to determine E(p,C),the maximum size of a connected graph G with the given vertex number p and cutwidth C. This paper presents some results on this problem.
文摘The circular chromatic number and the fractional chromatic number are two generalizations of the ordinary chromatic number of a graph. A graph is called star extremal if its fractional chromatic number equals to its circular chromatic number (also known as the star chromatic number). This paper studies the star extremality of the circulant graphs whose generating sets are of the form {±1,±k} .
