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A note on vertex pair sum $k$-zero ring labeling
Antony Sanoj Jerome,K.R. Santhosh Kumar,T.J. Rajesh Kumar 한국전산응용수학회 2024 Journal of applied mathematics & informatics Vol.42 No.2
Let $G=(V,E)$ be a graph with $p$-vertices and $q$-edges and let $R^{\circ}$ be a finite zero ring of order $n$. An injective function $f:V(G)\to \{r_1,r_2,\ldots,r_k\}$, where $r_i\in R^\circ$ is called vertex pair sum $k$-zero ring labeling, if it is possible to label the vertices $x\in V$ with distinct labels from $R^{\circ}$ such that each edge $e=uv$ is labeled with $f(e=uv)=[f(u)+f(v)] \pmod n$ and the edge labels are distinct. A graph admits such labeling is called vertex pair sum $k$-zero ring graph. The minimum value of positive integer $k$ for a graph $G$ which admits a vertex pair sum $k$-zero ring labeling is called the vertex pair sum $k$-zero ring index denoted by $\psi_{pz}(G)$. In this paper, we defined the vertex pair sum $k$-zero ring labeling and applied to some graphs.
The split and non-split tree $(D,C)$-number of a graph
P.A. Safeer,A. Sadiquali,K.R. Santhosh Kumar 한국전산응용수학회 2024 Journal of applied mathematics & informatics Vol.42 No.3
In this paper, we introduce the concept of split and non-split tree ($D,C$)- set of a connected graph $G$ and its associated color variable, namely split tree $(D,C)$ number and non-split tree $(D,C)$ number of $G$. A subset $S\subseteq V$ of vertices in $G$ is said to be a split tree $(D,C)$ set of $G$ if $S$ is a tree $(D,C)$ set and $\langle V-S\rangle$ is disconnected. The minimum size of the split tree $(D,C)$ set of $G$ is the split tree $(D,C)$ number of $G$, $\gamma_{\chi_{ST}}(G)=\min\{\vert S\vert:S\;\mbox{is a split tree}\; (D,C)\;\mbox{set}\}.$ A subset $S\subseteq V$ of vertices of $G$ is said to be a non-split tree $(D,C)$ set of $G$ if $S$ is a tree $(D,C)$ set and $\langle V-S\rangle$ is connected and non-split tree $(D,C)$ number of $G$ is $\gamma_{\chi_{NST}}(G)=\min\{\vert S\vert: S\; \mbox{is a non-split tree }\; (D,C)\;\mbox{set of $G$}\}.$ The split and non-split tree $(D,C)$ number of some standard graphs and its compliments are identified.