Maria Aguieiras A. de Freitas

Spectra of graphs obtained by a generalization of the join graph operation

Abstract Taking a Fiedler’s result on the spectrum of a matrix formed from two symmetric matrices as a motivation, a more general result is deduced and applied to the determination of adjacency and Laplacian spectra of graphs obtained by a generalized join graph operation on families of graphs (regular in the case of adjacency spectra and arbitrary in the case of Laplacian spectra). Some additional consequences are explored, namely regarding the largest eigenvalue and algebraic connectivity.

A note on a conjecture for the distance Laplacian matrix

In this note, the graphs of order n having the largest distance Laplacian eigenvalue of multiplicity n− 2 are characterized. In particular, it is shown that if the largest eigenvalue of the distance Laplacian matrix of a connected graph G of order n has multiplicity n − 2, then G = Sn or G = Kp,p, where n = 2p. This resolves a conjecture proposed by M. Aouchiche and P. Hansen in [M. Aouchiche and P. Hansen. A Laplacian for the distance matrix of a graph. Czechoslovak Mathematical Journal, 64(3):751–761, 2014.]. Moreover, it is proved that if G has P5 as an induced subgraph then the multiplicity of the largest eigenvalue of the distance Laplacian matrix of G is less than n− 3.

Split non-threshold Laplacian integral graphs

The aim of this article is to answer a question posed by Merris in European Journal of Combinatorics, 24 (2003) pp. 413 − 430, about the possibility of finding split non-threshold graphs that are Laplacian integral, i.e. graphs for which the eigenvalues of the corresponding Laplacian matrix are integers. Using Kronecker products, balanced incomplete block designs, and solutions to certain Diophantine equations, we show how to build infinite families of these graphs.

On Q-spectral integral variation

Abstract Let G be a connected graph with two nonadjacent vertices and G ′ be the graph constructed from G by adding an edge between them. It is known that the trace of Q ′ is 2 plus the trace of Q, where Q and Q ′ are the signless Laplacian matrices of G and G ′ , respectively. Hence, the sum of the eigenvalues of Q ′ is the sum of the eigenvalues of Q plus 2. Since none of the eigenvalues of Q can decrease if an edge is added to G, it is said that Q-spectral integral variation occurs when either only one Q-eigenvalue is increased by 2, or when two Q-eigenvalues are increased by 1 one each. In this article we give necessary and sufficient conditions for the occurrence of Q-spectral integral variation only in two places, as the first case never occurs.