Eigenvalue location for chain graphs

Eigenvalue inequalities for graphs and convex subgraphs

For an induced subgraph S of a graph, we show that its Neumann eigenvalue λS can be lower-bounded by using the heat kernel Ht(x, y) of the subgraph. Namely, λS ≥ 1 2t ∑ x∈S inf y∈S Ht(x, y) √ dx √ dy where dx denotes the degree of the vertex x. In particular, we derive lower bounds of eigenvalues for convex subgraphs which consist of lattice points in an d-dimensional Riemannian manifolds M wit...

Eigenvalue Bracketing for Discrete and Metric Graphs

We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph spectrum (also in the “exceptional” values of the metric graph corresponding to the Dirichlet spectrum) we carry over these estimates from the metric graph La...

Eigenvalue spectra for Gaussian semilinear polymer graphs

Graphs with reciprocal eigenvalue properties

In this paper, only simple graphs are considered. A graphG is nonsingular if its adjacency matrix A(G) is nonsingular. A nonsingular graph G satisfies reciprocal eigenvalue property (property R) if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and G satisfies strong reciprocal eigenvalue property (property SR) if the reciprocal of each eigenvalue o...

Eigenvalue Multiplicity in Cubic Graphs

Let G be a connected cubic graph of order n with μ as an eigenvalue of multiplicity k. We show that (i) if μ 6∈ {−1, 0} then k ≤ 12n, with equality if and only if μ = 1 and G is the Petersen graph; (ii) If μ = −1 then k ≤ 12n + 1, with equality if and only if G = K4; (iii) If μ = 0 then k ≤ 12n+ 1, with equality if and only if G = 2K3. AMS Classification: 05C50