I. Blessy, I. Gnanaselvi

Some Results on Square Prime Cordial Graphs

Introduction

Some results on square prime cordial graphs. Explore Square Prime Cordial (SPC) graphs and their labeling. This paper presents results on SPC labeling for broom graphs (Bn,m) and Kl,m * Kl,n graphs.

Abstract

Let G = (p, q) be a graph with p vertices and q edges. A SPC labeling of a graph G with vertex collection V(G) is a bijection a: V(G) --> {0, 1, 2,...., p-1} to the extent that its induced binary edge labeling function a* : E(G) ---> {0, 1} is defined by a*(uv) = 0 if [((a(u))2, (a(v))2) + 2d(u, v)] = 0 (mod 2); 1 if [((a(u))2, a(v))2) + 2d(u, v)] = 1 (mod 2). The difference between the edges designated as 0, denoted by ea(0) and the edges desinated as 1, denotd by ea(1) is atmost 1. A graph which follows SPC labeling is referred to as SPC graph. This paper elucidates that broom graph Bn, m and Kl, m * Kl, n admits SPC Labeling and some Results.

Review

This paper introduces a novel graph labeling scheme termed "Square Prime Cordial (SPC) labeling," aiming to classify graphs that admit such a labeling. The core concept involves assigning distinct integers from 0 to p-1 to the vertices of a graph G, and subsequently defining an induced binary edge labeling based on a specific arithmetic condition involving the squares of vertex labels and the distance between vertices. The goal, as implied by the "cordial" aspect, is to achieve a balanced distribution of 0-labeled and 1-labeled edges, where the difference between their counts is at most one. The abstract states that the paper demonstrates that broom graphs $B_{n,m}$ and a product of complete bipartite graphs $K_{l,m} * K_{l,n}$ are SPC graphs. Upon closer examination of the abstract, several critical issues regarding the clarity and mathematical rigor of the SPC labeling definition emerge. The notation "((a(u))2, (a(v))2)" is highly ambiguous; it could denote a pair, a greatest common divisor, or perhaps an implicit product, none of which are explicitly defined. More significantly, the term "2d(u, v)" in the labeling condition "[((a(u))2, (a(v))2) + 2d(u, v)] = 0 (mod 2)" is mathematically redundant. Since 2d(u, v) will always be an even number (as d(u,v) is an integer distance), its contribution modulo 2 is always 0. This means the edge labeling condition simplifies entirely to the parity of "((a(u))2, (a(v))2)", rendering the distance term irrelevant to the labeling outcome. This redundancy suggests a fundamental flaw in the proposed definition, or at least a lack of precision in its presentation within the abstract. Furthermore, a likely typographical error exists in the second part of the definition, where "(a(v))2)" should probably be "((a(v))^2)". In conclusion, while the introduction of new graph labeling schemes is a valuable contribution to graph theory, the abstract for "Some Results on Square Prime Cordial Graphs" presents a definition that is both ambiguous and contains a mathematically redundant term. The lack of clarity around the "((X)2, (Y)2)" notation and the simplification of the labeling condition due to the `2d(u,v)` term raise significant concerns about the novelty and well-foundedness of the proposed SPC labeling. For the full paper to be considered, these definitional issues must be thoroughly addressed, clarified, and rigorously justified. Without a precise and non-redundant definition, the results regarding broom graphs and $K_{l,m} * K_{l,n}$ cannot be properly evaluated or understood.

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