SAMS Subject Classification Number: 6

Let \(G\) be a connected graph. The diameter and Wiener index of \(G\) are defined as the largest and the sum, respectively, of the distances between all pairs of vertices of \(G\). The average eccentricity of \(G\) is the arithmetic mean of the eccentricities of the vertices of \(G\), where the eccentricity of a vertex \(v\) is defined as the distance between \(v\) and a vertex farthest from \(v\).

In this talk we present bounds on these distance measures for graphs of girth \(6\) in terms of order, minimum degree and maximum degree. We further show that, in a sense, our bounds are best possible. In this context a natural generalisation of cages arises, which is interesting in its own right. The cage problem considers the minimum order of regular graphs of prescribed degree and girth. In our generalisation we consider not regular graphs, but prescribe minimum and maximum degree.