Average distance and vertex‐connectivity

Abstract

Abstract:The average distance (G) of a connected graph G of order n is the average of the distances between all pairs of vertices of G, i.e., y), where V(G) denotes the vertex set of G and d G (x, y) is the distance between x and y. We prove that if G is a -vertex-connected graph, ≥ 3 an odd integer, of order n, then (G) ≤ n/2( +1)+O(1). Our bound is shown to be best possible and our results, apart from answering a question of Plesník [J Graph Theory 8 (1984), 1--24], Favaron et al. [Networks 19 (1989), 493--504], can be used to deduce a theorem that is essentially equivalent to a theorem by Egawa and Inoue [Ars Combin 51 (1999), 89--95] on the radius of a -vertex-connected graph of given order, where is odd. ᭧

Proof of a Conjecture on the Wiener Index of Eulerian Graphs

“…The proof of Theorem 2 given in the present paper suggest that the situation is no different for Eulerian graphs. We note that Plesník's result on 2-connected graphs was, asymptotically, extended to k-connected graphs in [8].…”

Counting the numbers of paths of all lengths in dendrimers and its applications

“…For example of the studies of average distance of graphs, Fajtlowicz and Waller [11] established the inequality between the average distance and the independence number in their classical paper since 1986 that α(G) ≥ µ(G) − 1 for every connected graph G. Chung [7] improved this bound to bt α(G) ≥ µ(G) and further characterized that the equality holds if and only if G is a complete graph. For more studies of the average distance of graphs see [8,13,27] for example.…”

The maximum Wiener index of maximal planar graphs

“…which is attained only by a path (Dankelmann et al 2008a;Plesník 1984;Lovász 1979). Many sharp or asymptotically sharp bounds on W (G) in terms of other graph parameters are known, for instance, minimum degree (Beezer et al 2001;Dankelmann and Entringer 2000;Kouider and Winkler 1997), connectivity (Dankelmann et al 2009;Favaron et al 1989), edge-connectivity (Dankelmann et al 2008b, a) and maximum degree (Fischermann et al 2002). For finding more details in mathematical aspect of Wiener index, see also results (Das and Nadjafi-Arani 2017;Gutman et al 2014;Klavžar and Nadjafi-Arani 2014;Knor et al , 2016Li et al 2018;Mukwembi and Vetrík 2014;Wagner et al 2009;Wagner 2006;Wang and Yu 2006).…”