An $\alpha$-triangle eigenvector centrality of graphs

Centrality represents a fundamental research field in complex network analysis, where centrality measures identify important vertices within networks. Over the years, researchers have developed diverse centrality measures from varied perspectives. This paper proposes an $\alpha$-triangle eigenvector centrality ($\alpha$TEC), which is a global centrality measure based on both edge and triangle structures. It can dynamically adjust the influence of edges and triangles through a parameter $\alpha$ ($\alpha \in (0,1]$). The centrality scores for vertices are defined as the eigenvector corresponding to the spectral radius of a nonnegative tensor. By the Perron-Frobenius theorem, $\alpha$TEC guarantees unique positive centrality scores for all vertices in connected graphs. Numerical experiments on synthetic and real world networks demonstrate that $\alpha$TEC effectively identifies the vertex's structural positioning within graphs. As $\alpha$ increases (decreases), the centrality rankings reflect a stronger (weaker) contribution from edge structure and a weaker (stronger) contribution from triangle structure. Furthermore, we experimentally prove that vertices with higher $\alpha$TEC rankings have a greater impact on network connectivity.