Enzo Maria Li Marzi

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 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.

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.

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.

Schwenk‐Like Formulas for Weighted Digraphs

Recently the study of the spectrum of weighted (di)graphs has attracted the interest of many researchers. Here we express the characteristic polynomial of any (square) matrix A in terms of the determinant of the Coates graph of the matrix B = xI−A. By doing so we are able to generalize the well‐known Schwenk’s formulas for simple graphs to weighted digraphs.

Signed line graphs with least eigenvalue - 2

We use star complement technique to construct a basis for - 2 of signed line graphs using their root signed graphs. In other words, we offer a generalization of the corresponding results known in the literature for (unsigned) graphs in the context of line graphs and generalized line graphs.

Perfect 7-cycle systems

Perfect m-cycle systems are defined. In this paper, it is proven that the class of perfect 7-cycle systems is a variety of quasigroups, and a defining set of identities for this variety is given.