Bounds in radial Moore graphs of diameter 3

TL;DR

This paper investigates bounds on the number of central vertices and the status in radial Moore graphs of diameter 3, proposing a family of graphs with maximum status and contributing to understanding their structural properties.

Contribution

It establishes upper bounds for central vertices and status in radial Moore graphs of diameter 3, and introduces a conjectured maximum status family.

Findings

Derived upper bounds for central vertices and status

Presented a family of radial Moore graphs with conjectured maximum status

Connected status measure to Moore graph approximations

Abstract

Radial Moore graphs are approximations of Moore graphs that preserve the distance-preserving spanning tree for its central vertices. One way to classify their resemblance with a Moore graph is the status measure. The status of a graph is defined as the sum of the distances of all pairs of ordered vertices and equals twice the Wiener index. In this paper we study upper bounds for both the maximum number of central vertices and the status of radial Moore graphs. Finally, we present a family of radial Moore graphs of diameter $3$ that is conjectured to have maximum status.