SKOLEM MEAN LIKE LABELING FOR FOUR-STAR GRAPHS: A COMPLETE CHARACTERISATION WITH NETWORK APPLICATIONS

Abstract

Graph labeling is a vibrant area of combinatorics with established connections to network design, coding theory, and cryptography. Among the many mean-type labeling schemes in the literature, Skolem Mean Like Labeling (SMLL) is distinguished by using vertex labels drawn from {1, 2, ..., p} (where p = q+1) and requiring that the induced ceiling-mean edge labels cover {2, 3…, p} exactly, omitting only the label 1. This shift makes SMLL especially natural for hierarchical star-chain topologies.
In this paper we establish a complete characterisation of the four-star graph G = K₁, θ₁ ∧ K₁, θ₁ ∧ K₁, θ₂ ∧ K₁, θ₃, with θ₁ ≤ θ₂ < θ₃, as a Skolem mean like graph. We prove that G admits a Skolem Mean Like Labeling if and only if |θ₂ - θ₃| ≤ 2θ₁ + 4 and 2θ₁ + θ₂ - 4 ≤ θ₃ ≤ 2θ₁ + θ₂ + 4. Sufficiency is shown through nine explicit constructions; necessity follows from an exhaustive contradiction argument. A master comparison table consolidates all nine cases. A concrete application to cloud server–client architectures is also presented.