Bell's Degree Variance and Degree Deviation in Graphs: Analyzing Optimal Graphs Based on These Irregularity Measures

An irregularity measure (IM) of a connected graph G is defined as a non-negative graph invariant that satisfies the condition‎ ‎IM(G) = ۰ if and only if G is a regular graph‎. ‎Among the prominent degree-based irregularity measures‎ ‎are Bell's degree variance, denoted as Var_B(G), and degree deviation‎, represented as S(G). Specifically, they are defined by the equations Var_B(G) = \frac{۱}{n} \sum_{i=۱}^{n} \left( d_i‎ - ‎\frac{۲m}{n} \right)^۲ and‎ ‎S(G)=\sum_{i=۱}^n \left|d_i‎- ‎\frac{۲m}{n}\right |‎, ‎where m is the number of edges and n is the number of vertices in G‎. ‎ This paper studies the properties of Bell's degree-variance and degree deviation for acyclic, unicyclic, and cactus graphs‎. ‎ Our analysis shows how these measures relate to graph topology and structure, influencing the overall irregularity. Additionally, we identify and analyze optimal graphs that minimize both irregularity measures, providing insights into their implications for network design, data structure optimization, and real-world applications. This study contributes to the understanding of graph irregularity and offers a framework for future research into irregularity measures across different classes of graphs.

کلیدواژه ها:

Irregularity‎ ، ‎Degree-variance‎ ، Degree deviation‎ ، ‎Optimal graphs‎ ، ‎Tree‎ ، ‎Unicyclic‎ ، ‎Cactus graphs