The Shrikhande graph is a strongly regular graph on 16 nodes. It is cospectral
with the rook graph 
, so neither of the two is determined
by spectrum.
The Shrikhande graph is the smallest distance-regular graph that is not distance-transitive
(Brouwer et al. 1989, p. 136). It has intersection
array 
.
The Shrikhande graph (denoted 
by Egawa 1981) can be constructed on a set
of vertices 
with 
with edges between
two vertices 
and 
iff 
, 
, and 
(mod 4) (Egawa 1981).
The Shrikhande graph has two generalized LCF notations of order 8, eleven of order 4, 53 of order 2, and 2900 of order 1. The graphs with LCF notations of orders four and eight are illustrated above.
The Shrikhande graph appears on the cover of the book Combinatorial Matrix Theory by Brualdi and Ryser (1991), illustrated above.
The plots above show the adjacency matrix, incidence matrix, and graph distance matrices for the Shrikhande graph.
It is an integral graph with graph spectrum 
.
The bipartite double graph of the Shrikhande graph is the Kummer graph.
The Doob graph 
is the graph given by the graph Cartesian product of 
copies of the Shrikhande graph with a Hamming
graph 
.
The Egawa graph 
is constructed as the graph Cartesian product
of 
copies of the Shrikhande graph and 
copies of a Hamming graph
(Egawa 1981).
The Shrikhande graph has graph genus 1 and its graph complement has graph genus 5 (E. Weisstein, Jan. 13, 2026). It is therefore a toroidal graph, as illustrated above.
The Shrikhande graph is implemented in the Wolfram Language as GraphData["ShrikhandeGraph"].
by the Parameters." J. Combin. Th., Ser. A 31,
108-125, 1981.House of Graphs. "Shrikhande Graph." 
Association Scheme." Ann. Math. Stat. 30,
781-798, 1959.van Dam, E. R. and Haemers, W. H. "Which
Graphs Are Determined by Their Spectrum?" Lin. Algebra Appl. 373,
139-162, 2003.