### Some Results on Walk Regular Graphs Which Are Cospectral to Its Complement

Mirko LEPOVIĆ
Source: Tokyo J. of Math. Volume 33, Number 1 (2010), 223-234.

#### Abstract

We say that a regular graph $G$ of order $n$ and degree $r\ge 1$ (which is not the complete graph) is strongly regular if there exist non-negative integers $\tau$ and $\theta$ such that $|S_i\cap S_j| = \tau$ for any two adjacent vertices $i$ and $j$, and $|S_i\cap S_j| = \theta$ for any two distinct non-adjacent vertices $i$ and $j$, where $S_k$ denotes the neighborhood of the vertex $k$. We say that a graph $G$ of order $n$ is walk regular if and only if its vertex deleted subgraphs $G_i = G\smallsetminus i$ are cospectral for $i = 1,2,\ldots ,n$. We here establish necessary and sufficient conditions under which a walk regular graph $G$ which is cospectral to its complement $\overline G$ is strongly regular.

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Primary Subjects: 05C50
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Permanent link to this document: http://projecteuclid.org/euclid.tjm/1279719588
Digital Object Identifier: doi:10.3836/tjm/1279719588
Mathematical Reviews number (MathSciNet): MR2682891

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