Spectra of Graphs

Spectral radius and fractional matchings in graphs

“…The eigenvalues of the quotient matrix interlace the eigenvalues of G. This partition is equitable if for each 1 ≤ i, j ≤ s, any vertex v ∈ V i has exactly b i,j neighbours in V j . In this case, the eigenvalues of the quotient matrix are eigenvalues of G and the spectral radius of the quotient matrix equals the spectral radius of G (see [2,5] for more details).…”

“…Also, if k is even, then the upper bound on 2|E(H)| can be replaced by k|V (H)| − 2, since even regular graphs cannot have a cut-edge, where an edge e in a connected graph G is a cut-edge of G if G − e is disconnected. By using the Interlacing Theorem ( [2,5], Lemma 1.6 [4]) and the fact that the spectral radius of a graph is at least its average degree, they proved that if G is k-regular and has no perfect matching, then λ 3 (G) ≥ min i∈{1,2,3} λ 1 (H i ) > min H∈H 2|E(H)|/|V (H)|, where H i ∈ H. The bound on λ 3 that Brouwer and Haemers found could be improved, since equality in the bound on the spectral radius in terms of the average degree holds only when graphs are regular. Later, Cioabǎ, Gregory, and Haemers [3] found the minimum of λ 1 (H) over all graphs H ∈ H. More generally, Cioabǎ and the author [4] determined connections between the eigenvalues of an l-edge-connected k-regular graph and its matching number when 1 ≤ l ≤ k − 2.…”

An interlacing approach for bounding the sum of Laplacian eigenvalues of graphs

“…Recall that the Laplacian matrix of G is L = D − A where D is the diagonal matrix of the vertex degrees and A is the adjacency matrix of G. Let us also recall the following basic result about interlacing (see [5], [2], or [1]). Theorem 1.1.…”

Adjacency preservers, symmetric matrices, and cores

“…In odd characteristic we will rely on the determinant of a quadratic form, which is defined by det(f ) := det(C). All nondegenerate quadratic forms sat- 2 for some invertible P . In the sequel we are concerned only whether a determinant of a given quadratic form is a square or not, so equivalent forms have the 'same' determinant in this sense.…”