Some degree and distance-based invariants of wreath products of graphs

TL;DR

This paper investigates degree and distance-based invariants of wreath product graphs, providing explicit formulas for Zagreb, Wiener, and Szeged indices, with applications in Chemical Graph Theory.

Contribution

It offers new explicit formulas for key invariants of wreath product graphs, linking them to the invariants of the factor graphs.

Findings

Explicit Zagreb index formula in terms of factor graphs' indices

Detailed distance analysis leading to Wiener index formula

Derived Szeged index formula for wreath product graphs

Abstract

The wreath product of graphs is a graph composition inspired by the notion of wreath product of groups, with interesting connections with Geometric Group Theory and Probability. This paper is devoted to the description of some degree and distance-based invariants, of large interest in Chemical Graph Theory, for a wreath product of graphs. An explicit formula is obtained for the Zagreb indices, in terms of the Zagreb indices of the factor graphs. A detailed analysis of distances in a wreath product is performed, allowing to describe the antipodal graph and to provide a formula for the Wiener index. Finally, a formula for the Szeged index is obtained. Several explicit examples are given.