The graphs which maximize and minimize respectively the largest eigenvalue over all unicyclic mixed graphs U on n vertices are determined. The unicyclic mixed graphs U with the largest eigenvalue λ 1(U)=n or λ 1(U...The graphs which maximize and minimize respectively the largest eigenvalue over all unicyclic mixed graphs U on n vertices are determined. The unicyclic mixed graphs U with the largest eigenvalue λ 1(U)=n or λ 1(U)∈(n,n+1] are characterized.展开更多
Very recently D.Vukicevic et al.[8]introduced a new topological index for a molecular graph G named Lanzhou index as∑_(u∈V(G))d_(u)d^(2)_(u),where d_(u)and d_(u)denote the degree of vertex u in G and in its compleme...Very recently D.Vukicevic et al.[8]introduced a new topological index for a molecular graph G named Lanzhou index as∑_(u∈V(G))d_(u)d^(2)_(u),where d_(u)and d_(u)denote the degree of vertex u in G and in its complement respectively.Lanzhou index Lz(G)can be expressed as(n-1)M_(1)(G)-F(G),where M_(1)(G)and F(G)denote the first Zagreb index and the forgotten index of G respectively,and n is the number of vertices in G.It turns out that Lanzhou index outperforms M_(1)(G)and F(G)in predicting the logarithm of the octanol-water partition coefficient for octane and nonane isomers.It was shown that stars and balanced double stars are the minimal and maximal trees for Lanzhou index respectively.In this paper,we determine the unicyclic graphs and the unicyclic chemical graphs with the minimum and maximum Lanzhou indices separately.展开更多
The Kirchhoff index Kf(G) of a graph G is defined to be the sum of the resistance distances between all pairs of vertices of G. In this paper, we develop a novel method for ordering the Kirchhoff indices of the comple...The Kirchhoff index Kf(G) of a graph G is defined to be the sum of the resistance distances between all pairs of vertices of G. In this paper, we develop a novel method for ordering the Kirchhoff indices of the complements of trees and unicyclic graphs. With this method, we determine the first five maximum values of Kf■ and the first four maximum values of Kf(ū),where ■ and ū are the complements of a tree T and unicyclic graph U, respectively.展开更多
The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. In this paper we show the expression of the nullity and nullity set of unicyclic graphs with n vertices and girth r, and characterize ...The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. In this paper we show the expression of the nullity and nullity set of unicyclic graphs with n vertices and girth r, and characterize the unicyclic graphs with extremal nullity.展开更多
In 2012, Gutman and Wagner proposed the concept of the matching energy of a graph and pointed out that its chemical applications can go back to the 1970s. The matching energy of a graph is defined as the sum of the ab...In 2012, Gutman and Wagner proposed the concept of the matching energy of a graph and pointed out that its chemical applications can go back to the 1970s. The matching energy of a graph is defined as the sum of the absolute values of the zeros of its matching polynomial. Let u and v be the non-isolated vertices of the graphs G and H with the same order, respectively. Let wi?be a non-isolated vertex of graph Gi?where i=1, 2, …, k. We use Gu(k)?(respectively, Hv(k)) to denote the graph which is the coalescence of G (respectively, H) and G1, G2,…, Gk?by identifying the vertices u (respectively, v) and w1, w2,…, wk. In this paper, we first present a new technique of directly comparing the matching energies of Gu(k)?and Hv(k), which can tackle some quasi-order incomparable problems. As the applications of the technique, then we can determine the unicyclic graphs with perfect matchings of order 2n with the first to the ninth smallest matching energies for all n≥211.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
A subset S of V is called a k-connected dominating set if S is a dominating set and the induced subgraph S has at most k components.The k-connected domination number γck（G） of G is the minimum cardinality taken ove...A subset S of V is called a k-connected dominating set if S is a dominating set and the induced subgraph S has at most k components.The k-connected domination number γck（G） of G is the minimum cardinality taken over all minimal k-connected dominating sets of G.In this paper,we characterize trees and unicyclic graphs with equal connected domination and 2-connected domination numbers.展开更多
This paper first elaborates the research situation and progress of Laplace characteristics and the eigenvalues value of graphs. The second is given an upper bound of characteristic value of a kind of special graph usi...This paper first elaborates the research situation and progress of Laplace characteristics and the eigenvalues value of graphs. The second is given an upper bound of characteristic value of a kind of special graph using the properties of similar matrices. At the same time, a new upper bound of Laplace characteristic values are given using properties of Laplace matrix and the similarity matrix, to improve the existing results. Then, we use the example of the upper bound of our results are more precise than some previous results. Finally the use Laplace non- zero eigenvalues of graph properties to give a bound expressions using the degree of square with a number of edges and the graph of the number, number of connected component expression map, it reflected the relationship between eigenvalues and the amount of Laplace.展开更多
基金Supported by the project item for young teachers of colleges and universities of Anhui province( 2 0 0 3jq1 0 1 ) and the project item of Anhui University for talents group construction
文摘The graphs which maximize and minimize respectively the largest eigenvalue over all unicyclic mixed graphs U on n vertices are determined. The unicyclic mixed graphs U with the largest eigenvalue λ 1(U)=n or λ 1(U)∈(n,n+1] are characterized.
基金Supported by the National Natural Science Foundation of China(11871256)the Chinese-Croatian bilateral project(7-22)。
文摘Very recently D.Vukicevic et al.[8]introduced a new topological index for a molecular graph G named Lanzhou index as∑_(u∈V(G))d_(u)d^(2)_(u),where d_(u)and d_(u)denote the degree of vertex u in G and in its complement respectively.Lanzhou index Lz(G)can be expressed as(n-1)M_(1)(G)-F(G),where M_(1)(G)and F(G)denote the first Zagreb index and the forgotten index of G respectively,and n is the number of vertices in G.It turns out that Lanzhou index outperforms M_(1)(G)and F(G)in predicting the logarithm of the octanol-water partition coefficient for octane and nonane isomers.It was shown that stars and balanced double stars are the minimal and maximal trees for Lanzhou index respectively.In this paper,we determine the unicyclic graphs and the unicyclic chemical graphs with the minimum and maximum Lanzhou indices separately.
基金Supported by the National Natural Science Foundation of China(11861011,11501133,11661010)。
文摘The Kirchhoff index Kf(G) of a graph G is defined to be the sum of the resistance distances between all pairs of vertices of G. In this paper, we develop a novel method for ordering the Kirchhoff indices of the complements of trees and unicyclic graphs. With this method, we determine the first five maximum values of Kf■ and the first four maximum values of Kf(ū),where ■ and ū are the complements of a tree T and unicyclic graph U, respectively.
文摘The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. In this paper we show the expression of the nullity and nullity set of unicyclic graphs with n vertices and girth r, and characterize the unicyclic graphs with extremal nullity.
文摘In 2012, Gutman and Wagner proposed the concept of the matching energy of a graph and pointed out that its chemical applications can go back to the 1970s. The matching energy of a graph is defined as the sum of the absolute values of the zeros of its matching polynomial. Let u and v be the non-isolated vertices of the graphs G and H with the same order, respectively. Let wi?be a non-isolated vertex of graph Gi?where i=1, 2, …, k. We use Gu(k)?(respectively, Hv(k)) to denote the graph which is the coalescence of G (respectively, H) and G1, G2,…, Gk?by identifying the vertices u (respectively, v) and w1, w2,…, wk. In this paper, we first present a new technique of directly comparing the matching energies of Gu(k)?and Hv(k), which can tackle some quasi-order incomparable problems. As the applications of the technique, then we can determine the unicyclic graphs with perfect matchings of order 2n with the first to the ninth smallest matching energies for all n≥211.
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.
文摘A subset S of V is called a k-connected dominating set if S is a dominating set and the induced subgraph S has at most k components.The k-connected domination number γck（G） of G is the minimum cardinality taken over all minimal k-connected dominating sets of G.In this paper,we characterize trees and unicyclic graphs with equal connected domination and 2-connected domination numbers.
文摘This paper first elaborates the research situation and progress of Laplace characteristics and the eigenvalues value of graphs. The second is given an upper bound of characteristic value of a kind of special graph using the properties of similar matrices. At the same time, a new upper bound of Laplace characteristic values are given using properties of Laplace matrix and the similarity matrix, to improve the existing results. Then, we use the example of the upper bound of our results are more precise than some previous results. Finally the use Laplace non- zero eigenvalues of graph properties to give a bound expressions using the degree of square with a number of edges and the graph of the number, number of connected component expression map, it reflected the relationship between eigenvalues and the amount of Laplace.
