On Constructing Edge Irregular q-Labelling of Several Book Graphs

Given a graph B = (V (B),E(B)) with the vertex set V (B) and the edge set E(B). A vertex labelling of a simple, connected, and undirected B, g : V (B) → {1, 2, . . . , q}
is called an edge irregular q-labelling on B if two arbitrary edges uv, u′v′ in B have different weights, where the weight of edge uv is defined as ωg(uv) = g(u) + g(v). Then the edge irregularity strength of B, denoted by es(B) is defined as the smallest integer q such that B can be labelled by an edge irregular q-labelling. Furthermore, the edge irregularity strength in the graph B with maximum degree Δ(B) satisfies es(B) ≥ max ⌈ |E(B)|+1 2 ⌉, Δ(B). In this study, we develop an edge irregular q-labelling and determine the exact value of the edge irregularity strength of triangular book graph Bp(C3), rectangular book graph Bp(C4), pentagonal book graph Bp(C5). We show that the exact value of the es(B) of Bp(C3), Bp(C4), Bp(C5) is equal to p + 2, ⌈ 3p+2 2 ⌉, ⌈ 4p+2 2 ⌉, respectively. We also investigate an edge irregular q-labelling and determine the exact value of the edge irregularity strength of book graph Bp(Cm) with additional (m − 2)p pendant edges and with additional p pendant edges for m ≥ 6. For any book graph Bp(Cm) for m ≥ 6, we obtain that ⌈ (m−1)p+2 2 ⌉ ≤ es(Bp(Cm)) ≤ ⌈mp+2 2 ⌉.