Structural and spectral properties of a family of deterministic recursive trees: Rigorous solutions

As one of the most significant models, the uniform recursive tree (URT) has found many applications in a variety of fields. In this paper, we study rigorously the structural features and spectral properties of the adjacency matrix for a family of deterministic uniform recursive trees (DURTs) that are deterministic versions of URT. Firstly, from the perspective of complex networks, we investigate analytically the main structural characteristics of DURTs, and obtain the accurate solutions for these properties, which include degree distribution, average path length, distribution of node betweenness, and degree correlations. Then we determine the complete eigenvalues and their corresponding eigenvectors of the adjacency matrix for DURTs. Our research may shed light in better understanding of the features for URT. Also, the analytical methods used here is capable of extending to many other deterministic networks, making the precise computation of their properties (especially the full spectrum characteristics) possible.