On Sum Neighbourhood Degree Centrality of Graphs

Centrality measures play a pivotal role in understanding the importance and influence of vertices within a graph. In this paper, we introduce a novel form of centrality measure called the sum neighbourhood degree centrality which is designed to quantify the structural importance of vertices in complex networks. By incorporating both its immediate connectivity and the structural characteristics of its neighbourhood, the proposed measure captures local and relational properties of a vertex more effectively than the existing centrality measures. We formally define the measure, establish its mathematical properties, and compare its behaviour with other measures using analytical results. This centrality is shown to provide improved discrimination among vertices and yields meaningful insights in to various classes of graphs and real-world networks. The main purpose of this paper is to build a theoretical property of sum neighbourhood degree centrality of vertices and sum neighbourhood degree centrality weight of graphs. Keywords: centrality measures; sum neighbourhood degree centrality; sum neighbourhood degree centrality weight.