Simon Mukwembi

A note on diameter and the degree sequence of a graph

Abstract In this note, we use a technique introduced by Dankelmann and Entringer [P. Dankelmann, R.C. Entringer, Average distance, minimum degree and spanning trees, J. Graph Theory 33 (2000) 1–13] to obtain a strengthening of an old classical theorem by Erdős, Pach, Pollack and Tuza [P. Erdős, J. Pach, R. Pollack, Z. Tuza, Radius, diameter, and minimum degree, J. Combin. Theory B 47 (1989) 73–79] on diameter and minimum degree. To be precise, we will prove that if G is a connected graph of order n and minimum degree δ , then its diameter does not exceed 3 ( n − t ) δ + 1 + O ( 1 ) , where t is the number of distinct terms of the degree sequence of G . The featured parameter, t , is attractive in nature and promising; more discoveries on it in relation to other graph parameters are envisaged.

On the eccentric connectivity index and Wiener index of a graph

Abstract Let G be a finite connected graph of order n and minimum degree δ. The eccentric connectivity index ξc (G) of G is defined as ξc (G) = Σv∊V (G) ecG (v)degG (v), where ecG (x) and degG (x) denote the eccentricity and degree of vertex x in G, respectively. We prove that the eccentric connectivity index of G satisfies , and construct graphs which asymptotically attain the bound. Our bound implies some known results by Došlić, Saheli & Vukičević [4], Morgan, Mukwembi & Swart [11], and Zhou & Du [16]. Further, we also determine upper bounds on the well-studied Wiener index in terms of the eccentric connectivity index.

Average Distance, Independence Number, and Spanning Trees

Let G be a connected graph of order n and independence number α. We prove that G has a spanning tree with average distance at most 23α, if ni¾?2α-1, and at most α+2, if n>2α-1. As a corollary, we obtain, for n sufficiently large, an asymptotically sharp upper bound on the average distance of G in terms of its independence number. This bound, apart from confirming and improving on a conjecture of Graffiti [8], is a strengthening of a theorem of Chung [1], and that of Fajtlowicz and Waller [8], on average distance and independence number of a graph.

Average Distance and Edge-Connectivity I

The average distance

WIENER INDEX OF TREES OF GIVEN ORDER AND DIAMETER AT MOST 6

\mu(G)

Upper bounds on the average eccentricity

of a connected graph

On spanning cycles, paths and trees

G

On the Gutman index and minimum degree

of order

Degree distance and minimum degree

n

Domination, radius, and minimum degree

is the average of the distances between all pairs of vertices of