Average eccentricity, minimum degree and maximum degree in graphs

Abstract

Let G be a connected finite graph with vertex set V(G). The eccentricity e(v) of a vertex v is the distance from v to a vertex farthest from v. The average eccentricity of G is defined as $\frac{1}{|V(G)|}\sum _{v \in V(G)}e(v)$. We show that the average eccentricity of a connected graph of order n, minimum degree $\delta$ and maximum degree $\Delta$ does not exceed $\frac{9}{4} \frac{n-\Delta -1}{\delta +1} \big ( 1 + \frac{\Delta -\delta }{3n} \big ) + 7$, and this bound is sharp apart from an additive constant. We give improved bounds for triangle-free graphs and for graphs not containing 4-cycles.