# Some Results on Walk Regular and Strongly Regular Graphs

### From NZJM

**New Zealand Journal of Mathematics**

**Vol.** 38, (2008), **Pages** 161-169

**M. LepoviÄ‡**

Department of Mathematics

Vuksanovica 32

34000, Kragujevac

Serbia (Yugoslavia)

mailto:lepovic@knez.uis.kg.ac.yu

**Abstract** We say that a regular graph *G* of order *n* and degree
(which is not the complete graph) is strongly regular if there exist
non-negative integers τ and θ such that for any two adjacent vertices *i* and *j*, and for any two distinct non-adjacent vertices *i* and *j*,
where *S*_{k} denotes the neighborhood of the vertex *k*. We prove
that a regular *G* is strongly regular if and only if its vertex
deleted subgraph has exactly two main
eigenvalues for . In particular, we show that

where μ_{1} and μ_{2} are the main eigenvalues of *G*_{i}.
Besides, we demonstrate that if *G* is a conference graph then *G*_{i}
is cospectral to *H*_{i}, where *H*_{i} is switching equivalent to *G*_{i}
with respect to *S*_{i}.

**Keywords** Walk regular graph, strongly regular graph, main eigenvalues,
conjugate adjacency matrices

**Classification** (MSC2000) 05 C 50