[Paper Review] The metric dimension of small distance-regular and strongly regular graphs

R. A. Bailey|arXiv (Cornell University)|Dec 17, 2013

Graph Labeling and Dimension Problems28 references18 citations

TL;DR

This paper presents the results of extensive computer calculations using GAP to determine the metric dimension of all small distance-regular and strongly regular graphs, including all such graphs on up to 34 vertices, low-valency distance-transitive graphs up to 100 vertices, and various other classes. The key contribution is a comprehensive database of metric dimensions for these graphs, verified independently for strongly regular graphs and extending known results for families like Johnson and Kneser graphs, Paley graphs, and Hadamard graphs.

ABSTRACT

A {\em resolving set} for a graph $Γ$ is a collection of vertices $S$, chosen so that for each vertex $v$, the list of distances from $v$ to the members of $S$ uniquely specifies $v$. The {\em metric dimension} of $Γ$ is the smallest size of a resolving set for $Γ$. A graph is {\em distance-regular} if, for any two vertices $u,v$ at each distance $i$, the number of neighbours of $v$ at each possible distance from $u$ (i.e. $i-1$, $i$ or $i+1$) depends only on the distance $i$, and not on the choice of vertices $u,v$. The class of distance-regular graphs includes all distance-transitive graphs and all strongly regular graphs. In this paper, we present the results of computer calculations which have found the metric dimension of all distance-regular graphs on up to 34 vertices, low-valency distance transitive graphs on up to 100 vertices, strongly regular graphs on up to 45 vertices, rank-$3$ strongly regular graphs on under 100 vertices, as well as certain other distance-regular graphs.

Motivation & Objective

• To compute the metric dimension of all distance-regular graphs on up to 34 vertices.
• To determine the metric dimension of low-valency distance-transitive graphs up to 100 vertices.
• To compute the metric dimension of all strongly regular graphs on up to 45 vertices, verifying prior results via an independent method.
• To extend the computation to rank-3 strongly regular graphs up to 100 vertices and specific families such as Hadamard graphs and Johnson graphs.
• To provide a comprehensive, verified database of metric dimensions for small, symmetric graphs using computational algebra systems.

Proposed method

• Utilization of the GAP computer algebra system to perform systematic computation of resolving sets for graphs.
• Implementation of algorithms to compute distances and verify resolving set properties in distance-regular and strongly regular graphs.
• Leveraging existing libraries, such as GAP’s library of primitive groups and Sloane’s Hadamard matrix database, to construct and analyze graphs.
• Applying the definition of metric dimension: finding the smallest set S such that all vertices have unique distance vectors to S.
• Using GRAPE for constructing and analyzing Hadamard graphs from known Hadamard matrices.
• Cross-validating results for strongly regular graphs against prior linear programming-based computations by Kratica et al.

Experimental results

Research questions

• RQ1 What is the metric dimension of all distance-regular graphs on up to 34 vertices?
• RQ2 How does the metric dimension behave in low-valency distance-transitive graphs up to 100 vertices?
• RQ3 What are the metric dimensions of all strongly regular graphs on up to 45 vertices, and how do they compare to prior results?
• RQ4 What is the metric dimension of rank-3 strongly regular graphs up to 100 vertices, including Paley and Johnson graphs?
• RQ5 What patterns or bounds emerge in the metric dimension of Hadamard graphs of order 4 to 20?

Key findings

• The metric dimension of the complete multipartite graph $K_{m_1, ext{...},m_r}$ is $\sum_{i=1}^r (m_i - 1)$, with a proof based on transversal selection and the pigeonhole principle.
• For Johnson graphs $J(n,2)$, the metric dimension is $n-2$ for $n \geq 4$, as seen in $J(9,2)$ with metric dimension 6 and $J(10,2)$ with metric dimension 7.
• The Higman–Sims graph on 100 vertices has a metric dimension at most 14, with the author suspecting this is the exact value.
• All Hadamard graphs of the same order have the same metric dimension, with $H(4)$ having metric dimension 4, $H(8)$ having 7, and $H(16)$ and $H(20)$ both having 10.
• The metric dimension of the Paley graph $P_{37}$ is 5, while $P_{41}$ has metric dimension 7, and larger Paley graphs like $P_{89}$ and $P_{97}$ have metric dimensions 8 and 8 respectively.
• The Gewirtz graph on 56 vertices has metric dimension 9, and the Hoffman–Singleton graph on 50 vertices has metric dimension 11.