Self 2-distance graphs

TL;DR

This paper classifies all finite simple self 2-distance graphs without certain cycles and proves that no cubic self 2-distance graphs exist, advancing understanding of these specialized graph structures.

Contribution

It fully characterizes specific self 2-distance graphs and establishes the non-existence of cubic instances, filling gaps in the classification of these graphs.

Findings

All such graphs with no 4-cycle, diamond, or triangle with a common vertex are determined.

No cubic self 2-distance graphs exist.

Abstract

All finite simple self $2$-distance graphs with no $4$-cycle, diamond, or triangles with a common vertex are determined. Utilizing these results, it is shown that there is no cubic self $2$-distance graphs.