Relationship Between Adjacency Matrices and Super $(a,d)$-edge-antimagic-total Labeling of Graphs

Abstract

Let $G=G(v,e)$ be a finite simple graph with $v$ vertices and $e$ edges. An $(a,d)$-$\text{edge-antimagic-vertex}$ (EAV) $\text{labeling}$ is a one-to-one mapping $f:V(G)\to \{1,2,\dots ,v\}$ such that for every edge $xy\in E(G)$, the edge-weight set $\{f(x)+f(y)\mid xy\in E(G)\}=\{a,a+d,a+2d,\dots ,a+(e-1)d\}$ for some positive integers $a$ and $d$. An $(a,d)$-$\text{edge-antimagic-total labeling}$ is a one-to-one mapping $f:V(G)\cup E(G)\to \{1,2,\dots ,v+e\}$ with the property that for every edge $xy\in E(G)$,$\{f(x)+f(y)+f(xy)\mid xy\in E(G)\}=\{a,a+d,a+2d,\dots ,a+(e-1)d\}.$ This labeling is called $\text{super }(a,d)\text{-edge-antimagic total labeling}$ if $f(V(G))=\{1,2,\dots ,v\}$. In this paper, we investigate the relationship between the adjacency matrix, $(a,d)$-edge-antimagic vertex labeling, and super $(a,d)$-edge-antimagic total labeling, and show how to manipulate this matrix to construct new $(a,d)$-edge-antimagic vertex labelings and new super $(a,d)$-edge-antimagic total graphs.