## Abstract

For distinct vertices *u* and *v* in a graph *G*, the *connectivity* between *u* and *v*, denoted \(\kappa _G(u,v)\), is the maximum number of internally disjoint *u*–*v* paths in *G*. The *average connectivity* of *G*, denoted \({\overline{\kappa }}(G),\) is the average of \(\kappa _G(u,v)\) taken over all unordered pairs of distinct vertices *u*, *v* of *G*. Analogously, for a directed graph *D*, the *connectivity* from *u* to *v*, denoted \(\kappa _D(u,v)\), is the maximum number of internally disjoint directed *u*–*v* paths in *D*. The *average connectivity* of *D*, denoted \({\overline{\kappa }}(D)\), is the average of \(\kappa _D(u,v)\) taken over all ordered pairs of distinct vertices *u*, *v* of *D*. An *orientation* of a graph *G* is a directed graph obtained by assigning a direction to every edge of *G*. For a graph *G*, let \({\overline{\kappa }}_{\max }(G)\) denote the maximum average connectivity among all orientations of *G*. In this paper we obtain bounds for \({\overline{\kappa }}_{\max }(G)\) and for the ratio \({\overline{\kappa }}_{\max }(G)/{\overline{\kappa }}(G)\) for all graphs *G* of a given order and in a given class of graphs. Whenever possible, we demonstrate sharpness of these bounds. This problem had previously been studied for trees. We focus on the classes of cubic 3-connected graphs, minimally 2-connected graphs, 2-trees, and maximal outerplanar graphs.

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## Notes

The family of trees described in Henning and Oellermann (2004) for which the lower bound is asymptotically sharp can be obtained as follows. For a given \(t \ge 1\), take three copies of \(K_{1,t}\), and identify a leaf from each copy of \(K_{1,t}\) in a single vertex (which will have degree 3). Let \(T_{3t+1}\) be such a tree. So if \(n=3t+1\), then \(\bar{\kappa }_{\max }(T_{3t+1}) = \frac{2n^2+14n-43}{9n(n-1)}\). We point out that it was incorrectly stated in Henning and Oellermann (2004) that for a tree

*T*of order \(n \ge 3\), we have \(\bar{\kappa }_{\max }(T) \ge \frac{2n^2+14n-43}{9n(n-1)}\). This inequality holds for \(n \ge 34\). However, for \(n<34\), we have$$\begin{aligned} \bar{\kappa }_{\max }(K_{1,n-1})&=\frac{\left\lfloor \frac{n-1}{2}\right\rfloor \left\lceil \frac{n-1}{2}\right\rceil +(n-1)}{n(n-1)}< \frac{2n^2+14n-43}{9n(n-1)}. \end{aligned}$$For \(n< 34\), the stars are in fact the extremal trees. That is, for \(n<34\), one can show that \(\bar{\kappa }_{\max }(K_{1,n-1}) \le \bar{\kappa }_{\max }(T)\) for every tree

*T*of order*n*.

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## Acknowledgements

The authors wish to thank the Banff International Research Station for their support of the focussed research group number 18frg233, “Measuring the Connectedness of Graphs and Digraphs”, during which most of the research for this manuscript was completed.

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Ortrud Oellermann supported by an NSERC Grant CANADA, Grant number RGPIN-2016-05237. Lucas Mol supported by an NSERC Grant CANADA, Grant number RGPIN-2021-04084.

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Casablanca, R.M., Dankelmann, P., Goddard, W. *et al.* The maximum average connectivity among all orientations of a graph.
*J Comb Optim* **43**, 543–570 (2022). https://doi.org/10.1007/s10878-021-00789-z

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DOI: https://doi.org/10.1007/s10878-021-00789-z

### Keywords

- Connectivity
- Average connectivity
- Orientations
- Cubic graphs
- Minimally 2-connected graphs
- Maximal outerplanar graphs