Pseudo-Euclidean representations of switching classes of Johnson and Hamming graphs with minimal dimension
Hiroshi Nozaki, Masashi Shinohara, Sho Suda
Abstract
This paper considers minimum-dimensional representations of graphs in pseudo-Euclidean spaces, where adjacency and non-adjacency relations are reflected in fixed scalar square values. A representation of a simple graph $(V,E)$ is a mapping $\varphi$ from the vertices to the pseudo-Euclidean space $\mathbb{R}^{p,q}$ such that $||\varphi(u)-\varphi(v)|| = a$ if $(u,v) \in E$, $b$ if $(u,v) \notin E$ and $u \ne v$, and $0$ if $u = v$, for some $a,b \in \mathbb{R}$, where $||\boldsymbol{x}|| = \langle\langle \boldsymbol{x}, \boldsymbol{x} \rangle\rangle = \sum_{i=1}^p x_i^2 - \sum_{j=1}^q x_{p+j}^2$ is the scalar square of $\boldsymbol{x}$ in $\mathbb{R}^{p,q}$. For a finite set $X$ in $\mathbb{R}^{p,q}$, define $A(X) = \{||\boldsymbol{x}-\boldsymbol{y}|| : \boldsymbol{x},\boldsymbol{y} \in X, \boldsymbol{x} \ne \boldsymbol{y} \}$. We call $X$ an $s$-indefinite-distance set if $|A(X)| = s$. An $s$-indefinite-distance set in $\mathbb{R}^{p,0} = \mathbb{R}^p$ is called an $s$-distance set. Graphs obtained from Seidel switching of a Johnson graph sometimes admit Euclidean or pseudo-Euclidean representations in low dimensions relative to the number of vertices. For example, Lisoněk (1997) obtained a largest 2-distance set in $\mathbb{R}^8$ and spherical 2-indefinite-distance sets in $\mathbb{R}^{p,1}$ for $p \ge 10$ from the switching classes of Johnson graphs. In this paper, we consider graphs in the switching classes of Johnson and Hamming graphs and classify those that admit representations in $\mathbb{R}^{p,q}$ with the smallest possible dimension $p+q$ among all graphs in the same class. This method recovers known results, such as the largest 2-(indefinite)-distance sets constructed by Lisoněk, and also provides a unified framework for determining the minimum dimension of representations for entire switching classes of strongly regular graphs.