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 $R$ be a finite commutative ring with unity and let $G = (V, E)$ be a simple graph. The zero-divisor graph denoted by $\Gamma(R)$ is a simple graph with vertex set as $R$ and two vertices $x, y \in R$ are adjacent in $\Gamma(R)$ if and only if $xy=0$. In \cite{SP}, the authors have studied the Laplacian eigen values of the graph $\Gamma(\Z_{n})$ and for distinct proper divisors $d_1, d_2, \cdots, d_k$ of $n$, they defined the sets as $\mathcal{A}_{d_i} = \{x \in \Z_{n} : (x, n) = d_i\}$, where $(x, n)$ denotes the greatest common divisor of $x$ and $n$. In this paper, we show that the sets $\mathcal{A}_{d_i}$, where $1 \leq i \leq k$ are actually the orbits of the group action: $Aut(\Gamma(R)) \times R \longrightarrow R$, where $Aut(\Gamma(R))$ denotes the automorphism group of $\Gamma(R)$. 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:$Aut(\Gamma(R)) \times R \longrightarrow R$