On Edge-Balance Index Sets of Flux Capacitors and $L$-products of Stars with Cycles

Abstract

Let $G$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$, and let ${\mathbb{Z}}_2=\{0,1\}$. Any edge labeling $f$ induces a partial vertex labeling $f^{+}:V(G)\to {\mathbb{Z}}_2$ assigning $0$ or $1$ to $f^{+}(v)$, $v$ being an element of $V(G)$, depending on whether there are more $0$-edges or $1$-edges incident with $v$, and no label is given to $f^{+}(v)$ otherwise. For each $i\in {\mathbb{Z}}_2$, let $v_f(i)=|\{v\in V(G):f^{+}(v)=i\}|$ and $e_f(i)=|\{e\in E(G):f(e)=i\}|$. An edge-labeling $f$ of $G$ is said to be edge-friendly if $|e_f(0)–e_f(1)|\le 1$. The edge-balance index set of the graph $G$ is defined as $\text{EBI}(G)=\{|v_f(0)–v_f(1)|:f\text{ is edge-friendly}\}$. In this paper, we investigate and present results concerning the edge-balance index sets of flux capacitors and $L$-products of stars with cycles.