Super Edge-Connectivity and Zeroth-Order Randić Index

Abstract Define the zeroth-order Randić index as R0(G)=∑x∈V(G)1dG(x), {R^0}\left( G \right) = \sum\nolimits_{x \in V\left( G \right)} {{1 \over {\sqrt {{d_G}} \left( x \right)}},} where dG(x) denotes the degree of the vertex x. In this paper, we present two sufficient conditions for graphs and triangle-free graphs, respectively, to be super edge-connected in terms of the zeroth-order Randić index.

[1]  Terunao Soneoka,et al.  Super Edge-Connectivity of Dense Digraphs and Graphs , 1992, Discret. Appl. Math..

[2]  Lutz Volkmann,et al.  Degree sequence conditions for maximally edge-connected graphs depending on the clique number , 2000, Discret. Math..

[3]  S. Unger.  Molecular Connectivity in Structure–activity Analysis , 1987 .

[4]  Francis T. Boesch,et al.  On unreliability polynomials and graph connectivity in reliable network synthesis , 1986, J. Graph Theory.

[5]  Lutz Volkmann,et al.  Sufficient conditions on the zeroth-order general Randić index for maximally edge-connected graphs , 2017, Discret. Appl. Math..

[6]  A. K. Kel'mans.  Asymptotic Formulas for the Probability of k-Connectedness of Random Graphs , 1973 .

[7]  Linda .LESNIAK.  Results on the edge-connectivity of graphs , 1974, Discret. Math..

[8]  P. Dankelmann,et al.  Inverse degree and edge-connectivity , 2009, Discret. Math..

[9]  Liming Xiong,et al.  Maximally edge-connected graphs and Zeroth-order general Randić index for 0 , 2014, Discret. Appl. Math..

[10]  Guojun Li,et al.  Maximally edge-connected graphs and Zeroth-order general Randić index for $$\alpha \le -1$$α≤-1 , 2016, J. Comb. Optim..

[11]  Miguel Angel Fiol.  On super-edge-connected digraphs and bipartite digraphs , 1992, J. Graph Theory.

[12]  Chengfu Qin,et al.  Inverse degree and super edge-connectivity , 2012, Int. J. Comput. Math..

[13]  J. Plesník,et al.  On equality of edge-connectivity and minimum degree of a graph , 1989 .

[14]  Lutz Volkmann,et al.  New sufficient conditions for equality of minimum degree and edge-connectivity , 1995, Ars Comb..

[15]  G. Chartrand.  A Graph-Theoretic Approach to a Communications Problem , 1966 .

[16]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .