Topological indices on self-similar graphs generated by groups

TL;DR

This paper derives exact formulas for various topological indices and graph invariants of Schreier graphs generated by tree automaton groups, advancing understanding of their combinatorial properties.

Contribution

It provides explicit formulas for diameters, matchings, Tutte polynomials, and indices of Schreier graphs from tree automaton groups, a novel comprehensive analysis.

Findings

Formulas for diameters and perfect matchings

Explicit Tutte polynomial expressions

Exact Wiener and Szeged indices

Abstract

In this paper, we determine precise formulas for the diameters, the number of perfect matchings, and the Tutte polynomials for an infinite family of finite graphs, namely the Schreier graphs of tree automaton groups, also called tree graph automata. This enables us to easily find the number of spanning trees, spanning forests, and an explicit form for the chromatic polynomials. In the second part of the paper, we provide the precise values for the Wiener and Szeged index of any tree graph automaton.