Preface: The congress “HyGraDe 2017” and a biographical note on Mario Gionfriddo

In this preface we give some details about the past Congress “HyGraDe 2017” and we briefly describe the academic career of Mario Gionfriddo, whose 70th birthday was celebrated during the congress

Spectral characterizations of signed lollipop graphs

Let Γ=(G,σ) be a signed graph, where G is the underlying simple graph and σ:E(G)→{+,-} is the sign function on the edges of G. In this paper we consider the spectral characterization problem extended to the adjacency matrix and Laplacian matrix of signed graphs. After giving some basic results, we study the spectral determination of signed lollipop graphs, and we show that any signed lollipop graph is determined by the spectrum of its Laplacian matrix

The International Congress on Hypergraphs, Graphs and Designs – HyGraDe 2017

Editoria

A lower bound for the first Zagreb index and its application

For a graph G, the first Zagreb index is defined as the sum of the squares of the vertices degrees. By investigating the connection between the first Zagreb index and the first three coefficients of the Laplacian characteristic polynomial, we give a lower bound for the first Zagreb index, and we determine all corresponding extremal graphs. By doing so, we generalize some known results, and, as an application, we use these results to study the Laplacian spectral determination of graphs with small first Zagreb index

On the spectral characterizations of 3-rose graphs

A rose graph with p petals (or p-rose graph) is a graph obtained by taking p cycles with just a vertex in common. In this paper, we prove that all 3-rose graphs, having at least one triangle, are determined by their Laplacian spectra and all 3-rose graphs axe determined by their signless Laplacian spectra

Spectral analysis of the wreath product of a complete graph with a cocktail party graph

Graph products and the corresponding spectra are often studied in the literature. A special attention has been given to the wreath product of two graphs, which is derived from the homonymous product of groups. Despite a general formula for the spectrum is also known, such a formula is far from giving an explicit spectrum of the compound graph. Here, we consider the latter product of a complete graph with a cocktail party graph, and by making use of the theory of circulant matrices we give a direct way to compute the (adjacency) eigenvalues

A switching method for constructing cospectral gain graphs

A gain graph over a group G, also referred to as G-gain graph, is a graph where an element of a group G, called gain, is assigned to each oriented edge, in such a way that the inverse element is associated with the opposite orientation. Gain graphs can be regarded as a generalization of signed graphs, among others. In this work, we show a new switching method to construct cospectral gain graphs. Some previous methods known for graph cospectrality follow as a corollary of our results

Computing the permanental polynomial of a matrix from a combinatorial viewpoint

Recently, in the book [A Combinatorial Approach to Matrix Theory and Its Applications, CRC Press (2009)] the authors proposed a combinatorial approach to matrix theory by means of graph theory. In fact, if A is a square matrix over any field, then it is possible to associate to A a weighted digraph Ga, called Coates digraph. Through Ga (hence by graph theory) it is possible to express and prove results given for the matrix theory. In this paper we express the permanental polynomial of any matrix A in terms of permanental polynomials of some digraphs related to Ga