Distinguishing number and distinguishing index of graphs from primary subgraphs

TL;DR

This paper investigates the distinguishing number and index of graphs formed by point-attaching primary subgraphs, with applications in chemistry, to understand symmetry-breaking labelings.

Contribution

It introduces new results on the distinguishing number and index for graphs constructed via point-attaching from primary subgraphs, relevant to chemical graph theory.

Findings

Derived bounds for distinguishing number and index

Characterized graphs with minimal distinguishing labelings

Applied results to chemical graph structures

Abstract

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. Let $G$ be a connected graph constructed from pairwise disjoint connected graphs $G_1,\ldots ,G_k$ by selecting a vertex of $G_1$, a vertex of $G_2$, and identify these two vertices. Then continue in this manner inductively. We say that $G$ is obtained by point-attaching from $G_1, \ldots ,G_k$ and that $G_i$'s are the primary subgraphs of $G$. In this paper, we consider some particular cases of these graphs that are of importance in chemistry and study their distinguishing number and index.