Vector Basis $\{(1,1,1,1), (1,1,1,0), (1,1,0,0), (1,0,0,0)\}$-Cordial Labeling of $D(T_{n})\odot mK_{1}$ and $D(Q_{n})\odot mK_{1}$

Abstract

Let $G$ be a $(p,q)$ graph. Let $V$ be an inner product space with basis $S$. We denote the inner product of the vectors $x$ and $y$ by $<x,y>$. Let $\phi: V(G) \rightarrow S$ be a function. For edge $uv$ assign the label $<\phi(u),\phi(v)>$. Then $\phi$ is called a vector basis $S$-cordial labeling of $G$ if $|\phi_{x}-\phi_{y}|\leq 1$ and $|\gamma_i-\gamma_j |\leq 1$ where $\phi_{x}$ denotes the number of vertices labeled with the vector $x$ and $\gamma_i$ denotes the number of edges labeled with the scalar $i$. A graph which admits a vector basis $S$-cordial labeling is called a vector basis $S$-cordial graph. In this paper, we prove that the graphs $D(T_{n})\odot mK_{1}$ and $D(Q_{n})\odot mK_{1}$ admit a vector basis \{(1,1,1,1),(1,1,1,0),(1,1,0,0),(1,0,0,0)\}-cordial.