Irregular Interval Valued Fuzzy Graphs

In this paper, we define irregular interval-valued fuzzy graphs and their various classifications. Size of regular interval-valued fuzzy graphs is derived. The relation between highly and neighbourly irregular interval-valued fuzzy graphs are established. Some basic theorems related to the stated graphs have also been presented.

💡 Research Summary

The paper introduces a novel classification within the framework of interval‑valued fuzzy graphs (IVFGs) by defining “irregular interval‑valued fuzzy graphs” and exploring their structural properties. An IVFG is a graph in which each vertex and each edge is associated with a pair of membership values – a lower bound μ⁻ and an upper bound μ⁺ – thereby representing uncertainty as an interval rather than a single scalar. This richer representation allows the modeling of real‑world networks where the strength of relationships is not precisely known but can be bounded.

Definition of Irregularity.
In classical (regular) IVFGs, every vertex v has the same total incident fuzzy weight, i.e., the sum of the interval values of all edges incident to v equals a constant k (both for lower and upper bounds). The authors relax this condition and call a graph irregular if there exists at least one vertex whose incident fuzzy weight differs from that of another vertex. This simple change captures non‑uniform connectivity patterns that appear in many practical systems, such as social networks with heterogeneous trust levels or sensor networks with varying link reliability.

Classification Scheme.
Three subclasses are distinguished:

1. Highly Irregular IVFGs – Every vertex has a distinct neighbourhood set N(v) and the interval values of all incident edges are pairwise different. Consequently, the graph exhibits minimal symmetry; each node plays a unique role.

2. Neighbourly Irregular IVFGs – Vertices may share the same neighbourhood, but the interval values on the edges connecting to those neighbours differ. This models situations where the underlying topology is homogeneous but the strength of each connection varies (e.g., users with the same friends but different levels of trust).

3. Near‑Regular IVFGs – The total incident fuzzy weight of each vertex lies within a bounded interval