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
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.
A. E. Brouwer, A. M. Cohen, A. Neumaier, Distance-regular graphs, Berlin: Springer-Verlag (1989),
386 p.
