Graph identification index

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.

Figures (5)

• Figure 1: Ranks and strings in the Petersen graph.
• Figure 2: Tripartitle graph $K_{1,\ 1,\ 2}$.
• Figure 3: Ranks and strings in $Y_5$.
• Figure 4: A caterpillar.
• Figure 5: Rank assignment for a symmetric caterpillar with $n=6$ and $L=4$.

Theorems & Definitions (17)

• Proposition 1.1
• proof
• Proposition 1.2
• proof
• Theorem 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• proof