Isometric Cycles and a Generalization of Moore Graphs

TL;DR

This paper investigates bounds on the size of graphs based on their isometric cycles, girth, and degree, introducing extremal graphs that generalize Moore graphs and exploring their structural properties.

Contribution

It establishes new bounds on graph order related to isometric cycles and girth, and characterizes extremal Moore-like graphs with specific girth and degree conditions.

Findings

Bound on graph order in terms of equator, girth, and degree

Existence and properties of Moore-like extremal graphs

Characterization of extremal graphs for specific girth and degree

Abstract

The equator of a graph is the length of a longest isometric cycle. We bound the order $n$ of a graph from below by its equator $q$, girth $g$ and minimum degree $\delta$ - and show that this bound is sharp when there exists a Moore graph with girth $g$ and minimum degree $\delta$. The extremal graphs that attain our bound give an analogue of Moore graphs. We prove that these extremal `Moore-like' graphs are regular, and that every one of their vertices is contained in some maximum length isometric cycle. We show that these extremal graphs have a highly structured partition that is unique, and easily derived from any of its maximum length isometric cycles. We characterize the extremal graphs with girth 3 and 4, and those with girth 5 and minimum degree 3. We also bound the order of $C_4$-free graphs with given equator and minimum degree, and show that this bound is nearly sharp. We conclude with some questions and conjectures further relating our extremal graphs to cages and Moore graphs.