Graph identification index

TL;DR

This paper introduces the ID-index, a new graph invariant based on vertex rank assignments and distance-based strings, providing bounds and exact values for various graph classes.

Contribution

It defines the ID-index, explores its properties, and computes it for several fundamental graph families, linking it to existing graph invariants.

Findings

Lower bound on the ID-index established

Exact ID-indices computed for paths, cycles, and complete graphs

Relations between ID-graphs and ID-indices analyzed

Abstract

We introduce the \emph{ID-index} of a finite simple connected graph. For a graph $G=(V,\ E)$ with diameter $d$, we let $f:V\longrightarrow \mathbb{R}$ assign \emph{ranks} to the vertices, then under $f$, each vertex $v$ gets a \emph{string}, which is a $d$-vector with the $i$-th coordinate being the sum of the ranks of the vertices that are of distance $i$ from $v$. The \emph{ID-index} of $G$, denoted by $IDI(G)$, is defined to be the minimum number $k$ for which there is an $f$ with $|f(V)|=k$, such that each vertex gets a distinct string under $f$. We present some relations between ID-graphs, which were defined by Chartrand, Kono, and Zhang, and their ID-indices; give a lower bound on the ID-index of a graph; and determine the ID-indices of paths, grids, cycles, prisms, complete graphs, some complete multipartite graphs, and some caterpillars.