Universal spectra of the disjoint union of regular graphs

A universal adjacency matrix of a graph $G$ with adjacency matrix $A$ is any matrix of the form $U = \alpha A + \beta I + \gamma J + \delta D$ with $\alpha \neq 0$, where $I$ is the identity matrix, $J$ is the all-ones matrix and $D$ is the diagonal matrix with the vertex degrees. In the case that $G$ is the disjoint union of regular graphs, we present an expression for the characteristic polynomials of the various universal adjacency matrices in terms of the characteristic polynomials of the adjacency matrices of the components. As a consequence we obtain a formula for the characteristic polynomial of the Seidel matrix of $G$, and the signless Laplacian of the complement of $G$ (i.e. the join of regular graphs)

Signless Laplacian energies of non-commuting graphs of finite groups and related results

The non-commuting graph of a non-abelian group $G$ with center $Z(G)$ is a simple undirected graph whose vertex set is $G\setminus Z(G)$ and two vertices $x, y$ are adjacent if $xy \ne yx$. In this study, we compute Signless Laplacian spectrum and Signless Laplacian energy of non-commuting graphs of finite groups. We obtain several conditions such that the non-commuting graph of $G$ is Q-integral and observe relations between energy, Signless Laplacian energy and Laplacian energy. In addition, we look into the energetic hyper- and hypo-properties of non-commuting graphs of finite groups. We also assess whether the same graphs are Q-hyperenergetic and L-hyperenergetic.Comment: 39 page

Q-spectral and L-spectral radius of subgroup graphs of dihedral group

Research on Q-spectral and L-spectral radius of graph has been attracted many attentions. In other hand, several graphs associated with group have been introduced. Based on the absence of research on Q-spectral and L-spectral radius of subgroup graph of dihedral group, we do this research. We compute Q-spectral and L-spectral radius of subgroup graph of dihedral group and their complement, for several normal subgroups. Q-spectrum and Lspectrum of these graphs are also observed and we conclude that all graphs we discussed in this paper are Q-integral dan L-integral

Spectral integral variation of signed graphs

We characterize when the spectral variation of the signed Laplacian matrices is integral after a new edge is added to a signed graph. As an application, for every fixed signed complete graph, we fully characterize the class of signed graphs to which one can recursively add new edges keeping spectral integral variation to make the signed complete graph.Comment: 18 pages, 0 figur

Integral complete r-partite graphs

A graph is called integral if all the eigenvalues of its adjacency matrix are integers. In this paper, we give a useful sufficient and necessary condition for complete r-partite graphs to be integral, from which we can construct infinite many new classes of such integral graphs. It is proved that the problem of finding such integral graphs is equivalent to the problem of solving some Diophantine equations. The discovery of these integral complete r-partite graphs is a new contribution to the search of such integral graphs. Finally, we propose several basic open problems for further study