Eigenvalue comparisons in graph theory

In this paper, we explore what happens when the same techniques are applied to the problem of estimating eigenvalues of the adjacency operator on finite graphs of bounded degree. In Theorem 7, we show how eigenvalues of the adjacency operator on a finite graph Γ may be bounded in terms of the biggest eigenvalues of the adjacency operator on “geodesic balls” in Γ. We find explicit bounds for the eigenvalues on the balls (Theorem 6), and in Theorem 8, we turn these into explicit estimates on certain eigenvalues of the adjacency operator on Γ.