In this paper, we study the signless Laplacian spectral radius of unicyclic graphs with prescribed number of pendant vertices or independence number. We also characterize the extremal graphs completely.

The signless Laplacian Q and edge-Laplacian S of a given graph may or not be invertible. Moore-Penrose inverses are studied. In particular, using the incidence matrix, we find combinatorial formulas for trees. Also, present odd unicyclic graphs.

Let B(n, g) be the class of bicyclic graphs on n vertices with girth g. In this paper, the graphs in B(n, g) with the largest signless Laplacian spectral radius are characterized.

In this paper, we investigate how the smallest signless Laplacian eigenvalue of a graph behaves when the graph is perturbed by deleting a vertex, subdividing edges or moving edges.

Let B(n, g) be the class of bicyclic graphs on n vertices with girth g. In this paper, the graphs in B(n, g) with the largest signless Laplacian spectral radius are characterized.

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc (G) of a graph G is the mean value of eccentricities of all vertices of G. The harmonic index H (G) of a graph G is defined as the sum of 2 di+dj over all edges vivj of G, where di denotes the degree of a vertex vi in G. In this paper, we determine the unique tree with minimum average...

The energy of a graph G is equal to the sum of absolute values of the eigenvalues of the adjacency matrix of G, whereas the Laplacian energy of a graph G is equal to the sum of the absolute value of the difference between the eigenvalues of the Laplacian matrix of G and the average degree of the vertices of G. Motivated by the work from Sharafdini an...