Extremal regular graphs under prescribed degree constraints

Determine whether a regular graph maximizes or minimizes the subpath number among graphs with prescribed maximum degree or prescribed minimum degree, and characterize the maximal and minimal graphs with respect to the subpath number within the class of r-regular graphs.

Background

The paper proves that adding an edge strictly increases the subpath number and consequently identifies trees and complete graphs as the minimum and maximum, respectively, among connected graphs with a fixed number of vertices. It also determines the extremal complete bipartite graph among bipartite graphs.

These monotonicity arguments do not directly resolve extremal problems when degree constraints are imposed. The authors specifically focus attention on regular graphs, with cubic graphs identified as the first nontrivial case after the already-understood 2-regular case.

References

Moreover, since the addition of an edge strictly increases the subpath number, an interesting question is whether a regular graph maximizes (resp. minimizes) the subpath number among all graphs with prescribed maximum degree (resp. minimum degree). This focuses one's attention to regular graphs and the question which graphs are maximal and which are minimal with respect to the subpath number in the class of $r$-regular graph.

— Invitation to the subpath number  (2503.00558 - Knor et al., 1 Mar 2025) in Section 1, Introduction; Section 6, Concluding remarks and further work

The maximum among $r$-regular graphs will be exponential, and so similar as in (and cited papers therein) one may wonder about the value $\limsup_{G \in _r} \sqrt[n]{pn(G)}$ where $_r$ is the family of $r$-regular graphs. This and the characterization of the extremal graphs in those classes, seem a very hard problem. In particular, the author doubts about the possibility for a general exact determination ofProb.~17.

— The number and average length of subpaths in graphs  (2608.23542 - Cambie, 24 Aug 2026) in Section 3, Results on the number of subpaths