On the Edge-Balance Index Sets of the Network Graph

Abstract

Let G be a graph with vertex set V(G) and edge set E(G), and let Z={0,1}. Any edge labeling f:E(G) → Z ₂ of a graph induces a vertex labeling f ⁺:V(G) → Z ₂. Defined by f ⁺:(V) = i if V is incident to more i-edges than (1-i)-edges, and f ⁺(v) is unlabeled if V is incident to an equal number of 0-edges than 1-edges. Denote by e _f (i) and v _f (i) the number of edges and vertices, labeled i .We call edge-friendly if |e _f (0) − e _f (1)| ≤ 1. Define the edge-balance index set of G as: {|v _f (0) − v _f (1)|: f is an edge friendly labeling of G}. In this paper, we will study the edge-balance index sets of the network graph $C_{2^m} \times P_{m_2} (m\geq2)$, and solve formula proof and graphic tectonic methods.