Vertex equitable labeling of double brooms.

Let G be a finite simple graph having vertex set V(G), edge set E(G), number of vertices |V(G)| = p, number of edges |E(G)| = q, and A = { 0 , 1 , 2 , ... , [ q 2 ] }. A vertex equitable labeling of G is a vertex labeling f: V(G) → A that, induces a bijective edge labeling f*: E(G) → {1, 2, 3,..., q} defined by f*(uv) = f(u) + f(v) such that |vf(a) - vf(b)| ≤ 1 for all a, b ∈ A, where vf(a) is the number of vertices v with f(v) = a for a ∈ A. A graph is called a vertex equitable graph if it admits a vertex equitable labeling. Let m and n be positif integers. A double broom B(m, n, n) is a tree obtained from a path of length m by joining each vertex of degree 1, vi, i ∈ {1, 2}, to n new vertices vi,1, vi,2,..., vi,n. In the literature, the vertex equitable labelings of many classes of graphs have been studied. In this paper we study the vertex equitable labeling of a double broom B(m, n, n). We find that then double broom is a vertex equitable graph. [ABSTRACT FROM AUTHOR]