On isomorphism between two constructions of antipodal distance-regular graphs

S. Goryainov

For the background and necessary definitions, we refer the reader to [2]. In [1], two new constructions of antipodal distance-regular graphs have been proposed. The author of [1] has left unsolved the question of whether these graphs were isomorphic to some known ones. Let us recall the main results of [1].

Let $\Gamma_B$ be a graph with vertex set $B = g^G \cup (g^{-1})^G$, where $g^G$ is a conjugacy class of elements of order $p$ of the group $G = PSL_2(p^n)$, and edge set $\{\{x,y\} | xy^{-1}\in B\}$, where $p$ is an odd prime number, $q = p^n \ge 5$.

Theorem 1. (Mukhametyanov) If $q \equiv 1 (4)$ then the graph $\widehat{\Gamma_B}$ is is distance-regular with intersection array $\{q,q-3,1;1,2,q\}$.

Let $\Gamma_J$ be a graph whose vertex set is the set of all elements of order $p$ of $G$ and edge set $\{\{x,y\}|xy^{-1}\in J\}$, where $J$ is a class of conjugate involutions of $G$.

Theorem 2. (Mukhametyanov) If $q \equiv 1,3(8)$ then the graph $\Gamma_J$ is disconnected and its connected components are two isomorphic distance-regular graphs with intersection array $\{q,q-3,1;1,2,q\}$.

The following theorem is [2, Proposition 12.5.3].

Theorem 3. (Mathon) Let $q = rm + 1$ be a prime power, where $r > 1$ and either $m$ is even or $q$ is a power of $2$. Let $V$ be a vector space of dimension $2$ over $F = F_q$ provided with a nondegenerate symplectic form $B$. Let $K$ be the subgroup of the multiplicative group $F^*$ of index $r$, and let $b \in F^*$. Then the graph $M(m,q)$ with vertex set $\{Kv\ |\ v \in V \setminus \{\overline{0}\}\}$, where $Ku \sim Kv$ if and only if $B(u,v) \in bK$ is distance-regular of diameter $3$ with $r(q+1)$ vertices and intersection array $\{q,q-m-1,1;1,m,q\}$.

In this work, we show the following result.

Theorem 4. The graph $\widehat{\Gamma_B}$ and each of the connected components of $\Gamma_J$ are isomorphic to the graph $M(2,q)$ with appropriate $q$.

Acknowledgement. The work is supported by RFBR (project 12-01-31098), and by the grant of the President of Russian Federation for young scientists (project MK-1719.2013.1).

References

1. I. T. Mukhametyanov, On distance-regular graphs on the set of of nonidentity $p$-elements of the group $L_2(p^n)$, Tr. Inst. Mat. Mekh. UrO RAN, 18, N3 (2012), 164–178.
2. A. E. Brouwer, A. M. Cohen, A. Neumaier, Distance-regular graphs, Berlin: Springer-Verlag (1989), 386 p.

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