5 research outputs found
The spectral radius of subgraphs of regular graphs
We give a bound on the spectral radius of subgraphs of regular graphs with
given order and diameter. We give a lower bound on the smallest eigenvalue of a
nonbipartite regular graph of given order and diameter
On the spectra and spectral radii of token graphs
Let be a graph on vertices. The -token graph (or symmetric -th
power) of , denoted by has as vertices the
-subsets of vertices from , and two vertices are adjacent when their
symmetric difference is a pair of adjacent vertices in . In particular,
is the Johnson graph , which is a distance-regular graph
used in coding theory. In this paper, we present some results concerning the
(adjacency and Laplacian) spectrum of in terms of the spectrum of .
For instance, when is walk-regular, an exact value for the spectral radius
(or maximum eigenvalue) of is obtained. When is
distance-regular, other eigenvalues of its -token graph are derived using
the theory of equitable partitions. A generalization of Aldous' spectral gap
conjecture (which is now a theorem) is proposed
The spectral radius of subgraphs of regular graphs
Let μ (G) and μmin (G) be the largest and smallest eigenvalues of the adjacency-matrix of a graph G. Our main results are: (i) Let G be a regular graph of order n and finite diameter D. If H is a proper subgraph of G, then μ(G)-μ(H) \u3e 1/nD. (ii) If G is a regular nonbipartite graph of order n and finite diameter D, then μ (G) +μmin (G) 1/nD