Topological indices on self-similar graphs generated by groups

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.

Figures (4)

• Figure 1: The path $P_3$ and the associated automaton $\mathcal{A}_{P_3}$.
• Figure 2: The Schreier graphs $\Gamma_1,\Gamma_2$ and $\Gamma_3$ of the tangled odometer.
• Figure 3: The Schreier graph $\Gamma_4$ of the tangled odometer.
• Figure 4: An $e$-cycle $C$ for $e = (1, 2)$ of size $2^4 = 16$ in the 5-th Schreier graph of the graph automaton given by a path on the three vertices 1, 2, 3, where 2 is the central vertex.

Theorems & Definitions (32)

• Definition 2.1
• Remark 2.2
• Proposition 3.1
• proof
• Remark 3.2
• Proposition 3.3
• proof
• Theorem 3.4
• proof
• Theorem 3.5