2 research outputs found
Some invariants related to threshold and chain graphs
Let G = (V, E) be a finite simple connected graph. We say a graph G realizes
a code of the type 0^s_1 1^t_1 0^s_2 1^t_2 ... 0^s_k1^t_k if and only if G can
obtained from the code by some rule. Some classes of graphs such as threshold
and chain graphs realizes a code of the above mentioned type. In this paper, we
develop some computationally feasible methods to determine some interesting
graph theoretical invariants. We present an efficient algorithm to determine
the metric dimension of threshold and chain graphs. We compute threshold
dimension and restricted threshold dimension of threshold graphs. We discuss
L(2, 1)-coloring of threshold and chain graphs. In fact, for every threshold
graph G, we establish a formula by which we can obtain the {\lambda}-chromatic
number of G. Finally, we provide an algorithm to compute the
{\lambda}-chromatic number of chain graphs
Realization of zero-divisor graphs of finite commutative rings as threshold graphs
Let be a finite commutative ring with unity and let be a
simple graph. The zero-divisor graph denoted by is a simple graph
with vertex set as and two vertices are adjacent in
if and only if . In \cite{SP}, the authors have studied the
Laplacian eigen values of the graph and for distinct proper
divisors of , they defined the sets as
, where denotes
the greatest common divisor of and . In this paper, we show that the
sets , where are actually the orbits of
the group action: , where
denotes the automorphism group of . We characterize
all finite commutative rings with unity of which zero-divisor graphs are not
\textit{threshold}. We study creation sequences, hyperenergeticity and
hypoenergeticity of zero-divisor graphs. We compute the Laplacian eigenvalues
of zero-divisor graphs realized by some classes of reduced and local rings. We
show that the Laplacian eigenvalues of zero-divisor graphs are the
representatives of orbits of the group action: