On Balance Index Sets of $L$-Products with Cycles and Complete Graphs

Abstract

Let $G$ be a graph with vertex set $V$ and edge set $E$. A labeling $f:V\to \{0,1\}$ induces a partial edge labeling $f^{\ast }:E\to \{0,1\}$ defined by $f^{\ast }(xy)=f(x)$ if and only if $f(x)=f(y)$ for each edge $xy\in E$. The balance index set of $G$, denoted $\text{BI}(G)$, is defined as $\{|f^{\ast -1}(0)–f^{\ast -1}(1)|:|f^{-1}(0)–f^{-1}(1)|\le 1\}$. In this paper, we study the balance index sets of graphs which are $L$-products with cycles and complete graphs.