Let G be a connected graph with vertex set V(G), order n=|V(G)|, minimum degree δ(G) and connectivity κ(G). The graph G is called maximally connected if κ(G) = δ(G). Define the inverse degree of G with no isolated...Let G be a connected graph with vertex set V(G), order n=|V(G)|, minimum degree δ(G) and connectivity κ(G). The graph G is called maximally connected if κ(G) = δ(G). Define the inverse degree of G with no isolated vertices as R(G) =Σv∈V(G)1/d(v) , where d(v) denotes the degree of the vertex v. We show that G is maximally connected if R(G) <1+2/δ + n-2δ+1/((n-1)(n-3)) .展开更多
In the paper, a counterexample of the Graffiti's conjecture (583) is given out whichproves the conjecture is false. And the best bounds of I(T) +a'(T) are got, where Tdenotes a free, I(T) denotes the inverse d...In the paper, a counterexample of the Graffiti's conjecture (583) is given out whichproves the conjecture is false. And the best bounds of I(T) +a'(T) are got, where Tdenotes a free, I(T) denotes the inverse degree of T and a'(T) is the matching of T.展开更多
The first Zagreb index M1 （G） is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index M2 （G） is equal to the sum of the products of the degrees of pairs of adjacent vertices of t...The first Zagreb index M1 （G） is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index M2 （G） is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph G. In this paper, we obtain lower and upper bounds on the first Zagreb index MI（G） of G in terms of the number of vertices （n）, number of edges （m）, maximum vertex degree （△）, and minimum vertex degree （δ）. Using this result, we find lower and upper bounds on M2（G）. Also, we present lower and upper bounds on M2（G） ＋ M2（G） in terms of n, m, △, and δ, where denotes the complement of G. Moreover, we determine the bounds on first Zagreb coindex MI（G） and second Zagreb coindex M2（G）. Finally, we give a relation between the first Zagreb index and the second Zagreb index of graph G.展开更多
基金Supported by the Natural Science Foundation of Xinjiang University（XYl10102） Sup- ported by the of NSFXJ（2010211A06）
文摘Let G be a connected graph with vertex set V(G), order n=|V(G)|, minimum degree δ(G) and connectivity κ(G). The graph G is called maximally connected if κ(G) = δ(G). Define the inverse degree of G with no isolated vertices as R(G) =Σv∈V(G)1/d(v) , where d(v) denotes the degree of the vertex v. We show that G is maximally connected if R(G) <1+2/δ + n-2δ+1/((n-1)(n-3)) .
文摘In the paper, a counterexample of the Graffiti's conjecture (583) is given out whichproves the conjecture is false. And the best bounds of I(T) +a'(T) are got, where Tdenotes a free, I(T) denotes the inverse degree of T and a'(T) is the matching of T.
基金Acknowledgements The authors are grateful to two anonymous referees for their careful reading of this paper and strict criticisms, and valuable comments on this paper, which have considerably improved the presentation of this paper. The first author was supported by the National Research Foundation funded by the Korean government （Grant No. 2013R1A1A2009341） the second author was supported by the National Natural Science Foundation of China （Grant No. 11201227）, the China Postdoctoral Science Foundation （2013M530253）, and the Natural Science Foundation of Jiangsu Province （BK20131357）.
文摘The first Zagreb index M1 （G） is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index M2 （G） is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph G. In this paper, we obtain lower and upper bounds on the first Zagreb index MI（G） of G in terms of the number of vertices （n）, number of edges （m）, maximum vertex degree （△）, and minimum vertex degree （δ）. Using this result, we find lower and upper bounds on M2（G）. Also, we present lower and upper bounds on M2（G） ＋ M2（G） in terms of n, m, △, and δ, where denotes the complement of G. Moreover, we determine the bounds on first Zagreb coindex MI（G） and second Zagreb coindex M2（G）. Finally, we give a relation between the first Zagreb index and the second Zagreb index of graph G.
