Given a vertex v of a graph G the second order degree of v denoted as d2（v） is defined as the number of vertices at distance 2 from v. In this paper we address the following question： What axe the sufficient condit...Given a vertex v of a graph G the second order degree of v denoted as d2（v） is defined as the number of vertices at distance 2 from v. In this paper we address the following question： What axe the sufficient conditions for a graph to have a vertex v such that d2（v） ≥ d（v）, where d（v） denotes the degree of v？ Among other results, every graph of minimum degree exactly 2, except four graphs, is shown to have a vertex of second order degree as large as its own degree. Moreover, every K4^--free graph or every maximal planar graph is shown to have a vertex v such that d2（v） ≥ d（v）. Other sufficient conditions on graphs for guaranteeing this property axe also proved.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 Ministry of Education and Science,Spainthe European Regional Development Fund (ERDF)under project MTM2008-06620-C03-02+2 种基金the Catalan Government under project 2009 SGR 1298CONACyTMxico under project 57371PAPIIT-UNAM IN104609-3
文摘Given a vertex v of a graph G the second order degree of v denoted as d2（v） is defined as the number of vertices at distance 2 from v. In this paper we address the following question： What axe the sufficient conditions for a graph to have a vertex v such that d2（v） ≥ d（v）, where d（v） denotes the degree of v？ Among other results, every graph of minimum degree exactly 2, except four graphs, is shown to have a vertex of second order degree as large as its own degree. Moreover, every K4^--free graph or every maximal planar graph is shown to have a vertex v such that d2（v） ≥ d（v）. Other sufficient conditions on graphs for guaranteeing this property axe also proved.
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.
