Francesco Belardo

On the spectral characterizations of ∞-graphs

A ~-graph is a graph consisting of two cycles with just a vertex in common. We first look for some invariants for cospectral graphs, then we introduce a new method to determine the degree sequence of cospectral mates of a graph. In this paper, we prove that all ~-graphs without triangles are determined by their Laplacian spectra and that all ~-graphs, with one exception, are determined by their signless Laplacian spectra. For the exception we determine all graphs that are cospectral (w.r.t. signless Laplacian spectrum) to it.

On the two largest Q-eigenvalues of graphs

In this paper, we first give an upper bound for the largest signless Laplacian eigenvalue of a graph and find all the extremal graphs. Secondly, we consider the second-largest signless Laplacian eigenvalue and we characterize the connected graphs whose second-largest signless Laplacian eigenvalue does not exceed 3. Furthermore, we give the signless Laplacian spectral characterization of the latter graphs. In particular, the well-known friendship graph is proved to be determined by the signless Laplacian spectrum.

On the index of caterpillars

The index of a graph is the largest eigenvalue of its adjacency matrix. Among the trees with a fixed order and diameter, a graph with the maximal index is a caterpillar. In the set of caterpillars with a fixed order and diameter, or with a fixed degree sequence, we identify those whose index is maximal.

Ordering graphs with index in the interval (2,2+5)

The index of a graph is the largest eigenvalue (or spectral radius) of its adjacency matrix. We consider the problem of ordering graphs by the index in the class of connected graphs with a fixed order n and index belonging to the interval (2,2+5). For any fixed n (provided that n is not too small), we order a significant portion of graphs whose indices are close to the end points of the above interval.

Laplacian spectral characterization of disjoint union of paths and cycles

The Laplacian spectrum of a graph consists of the eigenvalues (together with multiplicities) of the Laplacian matrix. In this article we determine, among the graphs consisting of disjoint unions of paths and cycles, those ones which are determined by the Laplacian spectrum. For the graphs, which are not determined by the Laplacian spectrum, we give the corresponding cospectral non-isomorphic graphs.

Signless Laplacian spectral characterization of line graphs of T-shape trees

A T-shape tree is a tree with exactly one vertex of maximum degree 3. The line graphs of the T-shape trees are triangles with a hanging path at each vertex. Let Ca,b,c be such a graph, where a, b and c are the lengths of the paths. In this paper, we show that line graphs of T-shape trees, with the sole exception of Ca,a,2a+1, are determined by the spectra of their signless Laplacian matrices. For the graph Ca,a,2a+1 we identify the unique non-isomorphic graph sharing the same signless Laplacian characteristic polynomial.

Signless Laplacian eigenvalues and circumference of graphs

In this paper, we investigate the relation between the Q-spectrum and the structure of G in terms of the circumference of G. Exploiting this relation, we give a novel necessary condition for a graph to be Hamiltonian by means of its Q-spectrum. We also determine the graphs with exactly one or two Q-eigenvalues greater than or equal to 2 and obtain all minimal forbidden subgraphs and maximal graphs, as induced subgraphs, with respect to the latter property.

On the Index of Necklaces

We consider the following two classes of simple graphs: open necklaces and closed necklaces, consisting of a finite number of cliques of fixed orders arranged in path-like pattern and cycle-like pattern, respectively. In these two classes we determine those graphs whose index (the largest eigenvalue of the adjacency matrix) is maximal.

Some results on the signless Laplacians of graphs

In this paper we give two results concerning the signless Laplacian spectra of simple graphs. Firstly, we give a combinatorial expression for the fourth coefficient of the (signless Laplacian) characteristic polynomial of a graph. Secondly, we consider limit points for the (signless Laplacian) eigenvalues and we prove that each non-negative real number is a limit point for (signless Laplacian) eigenvalue of graphs.

Trees with minimal index and diameter at most four

In this paper we consider the trees with fixed order n and diameter d@?4. Among these trees we identify those trees whose index is minimal.