In this paper, various modifications of a connected graph G are regarded as perturbations of its signless Laplacian matrix. Several results concerning the resulting changes to the signless Laplacian spectral radius of G are obtained by solving intermediate eigenvalue problems of the second type. AMS subject classifications: 05C50

Let G be a simple graph with adjacency matrix A (= AG). The eigenvalues and the spectrum of A are also called the eigenvalues and the spectrum of G, respectively. If we consider a matrix Q = D + A instead of A, where D is the diagonal matrix of vertex–degrees (in G), we get the signless Laplacian eigenvalues and the signless Laplacian spectrum, respectively. For short, the signless Laplacian ei...

Let $S(G)$ be the Seidel matrix of a graph $G$ of order $n$ and let $D_S(G)=diag(n-1-2d_1, n-1-2d_2,ldots, n-1-2d_n)$ be the diagonal matrix with $d_i$ denoting the degree of a vertex $v_i$ in $G$. The Seidel Laplacian matrix of $G$ is defined as $SL(G)=D_S(G)-S(G)$ and the Seidel signless Laplacian matrix as $SL^+(G)=D_S(G)+S(G)$. The Seidel signless Laplacian energy $E_{SL^+...

Abstract In this paper, we give the spectrum of a matrix by using the quotient matrix, then we apply this result to various matrices associated to a graph and a digraph, including adjacency matrix, (signless) Laplacian matrix, distance matrix, distance (signless) Laplacian matrix, to obtain some known and new results. Moreover, we propose some problems for further research. AMS Classification: ...

We consider the problem of determining the Q–integral graphs, i.e. the graphs with integral signless Laplacian spectrum. First, we determine some infinite series of such graphs having the other two spectra (the usual one and the Laplacian) integral. We also completely determine all (2, s)–semiregular bipartite graphs with integral signless Laplacian spectrum. Finally, we give some results conce...

For a connected graph G, we derive tight inequalities relating the smallest signless Laplacian eigenvalue to the largest normalised Laplacian eigenvalue. We investigate how vectors yielding small values of the Rayleigh quotient for the signless Laplacian matrix can be used to identify bipartite subgraphs. Our results are applied to some graphs with degree sequences approximately following a pow...

A graph is said to be determined by its signless Laplacian spectrum if there is no other non-isomorphic graph with the same spectrum. In this paper, it is shown that each starlike tree with maximum degree 4 is determined by its signless Laplacian spectrum.

Recall that, given a graph G, the matrix Q = D + A is called the signless Laplacian, where A is the adjacency matrix and D is the diagonal matrix of vertex degrees. The matrix L = D − A is known as the Laplacian of G. Graphs with the same spectrum of an associated matrix M are called cospectral graphs with respect to M , or M–cospectral graphs. A graph H cospectral with a graph G, but not isomo...

We propose a simple and natural definition for the Laplacian and the signless Laplacian tensors of a uniform hypergraph. We study their H-eigenvalues, i.e., H-eigenvalues with nonnegative H-eigenvectors, and H-eigenvalues, i.e., H-eigenvalues with positive H-eigenvectors. We show that each of the Laplacian tensor, the signless Laplacian tensor, and the adjacency tensor has at most one H-eigenva...

The investigation of the spectral distances of graphs that started in [3] (I. Jovanović, Z. Stanić, Spectral distances of graphs, Linear Algebra Appl., 436 (2012) 1425–1435.) is continued by defining Laplacian and signless Laplacian spectral distances and considering their relations to the spectral distances based on the adjacency matrix of graph. Some separate results concerning the defined di...