Journal of Prime Research in Mathematics

# Vertex equitable labeling for ladder and snake related graphs

A. Lourdusamy
Department of Mathematics, St.Xavier’s College, Palayamkottai-627002, India.
F. Patrick
Department of Mathematics, St.Xavier’s College, Palayamkottai-627002, India.

$$^{1}$$Corresponding Author: lourdusamy15@gmail.com

### Abstract

Let $$G$$ be a graph with p vertices and q edges and $$A = {0, 1, 2, · · · ,\frac{q}{2}}$$. A vertex labeling $$f : V (G) → A$$ induces an edge labeling $$f^∗$$ defined by $$f^∗ (uv) = f(u) + f(v)$$ for all edges $$uv$$. For $$a ∈ A$$, let $$v_f (a)$$ be the number of vertices $$v$$ with$$f(v) = a$$. A graph $$G$$ is said to be vertex equitable if there exists a vertex labeling f such that for all $$a$$ and $$b$$ in A, $$|v_f (a) − v_f (b)| ≤ 1$$ and the induced edge labels are $${1, 2, 3, · · · , q}$$. In this paper, we prove that triangular ladder $$T L_n, L_n ⊙ mK_1, Q_n ⊙ K_1, T L_n⊙K_1$$ and alternate triangular snake $$A(T_n)$$ are vertex equitable graphs.

#### Keywords:

Vertex equitable labeling, ladder, snake.