Some spectral properties of the non-backtracking matrix of a graph

• Published in: Linear algebra and its applications Vol. 618; pp. 37 - 57
• Main Authors: Glover, Cory, Kempton, Mark
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.06.2021; American Elsevier Company, Inc
• Subjects: Apexes; Eigenvalues; Eigenvalues and eigenvectors; Eigenvectors; Graph theory; Linear algebra; Lower bounds; Non-backtracking walks; Spectra; Spectral graph theory; Upper bounds; Non-backtracking walks; 05C50; Spectral graph theory; Eigenvalues and eigenvectors; 15A18
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2021.01.022

We investigate the spectrum of the non-backtracking matrix of a graph. In particular, we show how to obtain eigenvectors of the non-backtracking matrix in terms of eigenvectors of a smaller matrix. In doing so, we create a block diagonal decomposition of the non-backtracking matrix, more clearly expressing its eigenvalues. Furthermore, we find an expression for the eigenvalues of the non-backtracking matrix in terms of eigenvalues of the adjacency matrix, and use this to upper-bound the spectral radius of the non-backtracking matrix, and to give a lower bound on the spectrum. Specifically, an upper-bound on the spectral radius is found in terms of the number of nodes and edges of the graph. We also investigate properties of a graph that can be determined by the spectrum. Specifically, we prove that the number of components, the number of degree 1 vertices, and whether or not the graph is bipartite are all determined by the spectrum of the non-backtracking matrix.