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...

Nested split and double nested graphs (commonly named nested graphs) are considered. General statements regarding the signless Laplacian spectra are proven, and the nested graphs whose second largest signless Laplacian eigenvalue is bounded by a fixed integral constant are studied. Some sufficient conditions are provided and a procedure for classifying such graphs in particular cases is provide...

for a simple connected graph $g$ with $n$-vertices having laplacian eigenvalues‎ ‎$mu_1$‎, ‎$mu_2$‎, ‎$dots$‎, ‎$mu_{n-1}$‎, ‎$mu_n=0$‎, ‎and signless laplacian eigenvalues $q_1‎, ‎q_2,dots‎, ‎q_n$‎, ‎the laplacian-energy-like invariant($lel$) and the incidence energy ($ie$) of a graph $g$ are respectively defined as $lel(g)=sum_{i=1}^{n-1}sqrt{mu_i}$ and $ie(g)=sum_{i=1}^{n}sqrt{q_i}$‎. ‎in th...