The regular adjacency bounds are upper bounds and lower bounds on the vertex
counts of vertex-induced subgraphs
that are regular graphs in a strongly
regular graph. Let 
be a strongly regular
graph with parameters 
, and let 
. The regular adjacency polynomial
of 
is
 |
If 
is a vertex-induced subgraph of 
that is a regular graph
of vertex degree 
and has vertex count 
,
then 
for every 
(Evans 2023). Define
 |
If 
is nonempty, then 
and 
are respectively the regular adjacency upper
bound and lower bound. Every nonempty vertex-induced
subgraph that is a regular graph of vertex
degree 
has vertex count between these two integers.
If 
is empty, no such vertex-induced subgraph
exists, and the conventional upper bound and lower
bound are 0 and 
, respectively.
The regular adjacency upper bound is no greater than the floored Haemers regular upper bound.
For every fixed 
, Evans (2023) proved that it is strictly smaller for
infinitely many strongly regular graphs.
