On the Signless Laplacian ABC-Spectral Properties of a Graph

In the paper, we introduce the signless Laplacian $ABC$-matrix $\mathrm{Q}\text{̃}\left(G\right)=\overline{D}\left(G\right)+\text{Ã}\left(G\right)$, where $\overline{D}\left(G\right)$ is the diagonal matrix of $ABC$-degrees and $\text{Ã}\left(G\right)$ is the $ABC$-matrix of $G.$ The eigenvalues of the matrix $\mathrm{Q}\text{̃}\left(G\right)$ are the signless Laplacian $ABC$-eigenvalues of G. We give some basic properties of the matrix $\mathrm{Q}\text{̃}\left(G\right)$, which includes relating independence number and clique number with signless Laplacian $ABC$-eigenvalues. For bipartite graphs, we show that the signless Laplacian $ABC$-spectrum and the Laplacian $ABC$-spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian $ABC$-eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian $ABC$-eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix $\mathrm{Q}\text{̃}\left(G\right)-\frac{tr\left(\mathrm{Q}\text{̃}\right(G\left)\right)}{n}I$, called the signless Laplacian $ABC$-energy of G. We obtain some upper and lower bounds for signless Laplacian $ABC$-energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the $ABC$-energy with the signless Laplacian $ABC$-energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian $ABC$-energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same $ABC$-energy.