Non-Conjugate Graphs Associated With Finite Groups

Let
$G$
be a finite group and
$S$
be a non-empty subset of
$G$
comprising of the non-conjugate elements. In this study, we introduced the non-conjugate graph associated with
$G$
with a coinciding set of vertices, such that two distinct vertices
$x$
and
$y$
are adjacent only if
$x,y\in S$
. We then discussed some fundamental properties to ensure the algebraic and combinatorial structure of the graph. Afterward, we formulated the resolving set and resolving polynomial for a subclass of dicyclic groups.