ABSTRACT
Suppose A is an n×n-matrix of complex entries ai,j. Then it has n (not necessarily distinct) eigenvalues λi ∈ C, i ∈ [n]. The maximum modulus (absolute value) of these eigenvalues
ρ(A) := max i∈[n] |λi|
is called the spectral radius of A. This chapter is dedicated to connections between the spectral radius and the row and column sums of a matrix or its powers. In particular, we investigate the relations between the number of walks in a graph and the largest eigenvalue of its adjacency matrix.
