index and the diameter of a graph ∗

Let G be a graph on n vertices with r: = ⌊n/2 ⌋ and let λ1 � · · · � λn be adjacency eigenvalues of G. Then the Hückel energy of G, HE(G), is defined as r∑ 2 λi, if n = 2r; i=1 HE(G) = r∑ 2 λi + λr+1, if n = 2r + 1. i=1 The concept of Hückel energy was introduced by Coulson as it gives a good approximation for the π-electron energy of molecular graphs. We obtain two upper bounds and a lower bound for HE(G). When n is even, it is shown that equality holds in both upper bounds if and only if G is a strongly regular graph with parameters (n,k,λ,µ) = (4t 2 + 4t + 2, 2t 2 + 3t + 1, t 2 + 2t, t 2 + 2t + 1), for positive integer t. Furthermore, we will give an infinite family of these strongly regular graph whose construction was communicated by Willem Haemers to us. He attributes the construction to J.J. Seidel.

Il presente lavoro di tesi si inserisce nell’ambito di ricerca dell’Ottimizzazione Discreta e riguarda, in particolare, modelli e metodi per problemi di ottimizzazione combinatoria. L’argomentazione è incentrata su un problema di completamento di biclique su grafo bipartito, denominato k-Clustering minimum Biclique Completion. Il campo applicativo principale a cui si è fatto riferimento è quello delle telecomunicazioni e riguarda l’aggregazione di sessioni per trasmissioni multicast. L’obiettivo principale del lavoro consiste nel progettare ed implementare metodi di soluzione efficaci, sia euristici che esatti. Il primo passo corrisponde alla formalizzazione del problema in modelli matematici di programmazione intera. Dei vari modelli si sono studiate le proprietà per mettere in luce le difficoltà che sarebbero potute nascere nell’ottenere una soluzione ottima. I modelli sono stati sottoposti a test mediante un risolutore commerciale. Successivamente si è progettato una euristica di ricerca locale, basata su intorni variabili e ricerca nella regione

Recently Hansen and Vukičević [10] proved that the inequality M1/n ≤ M2/m, where M1 and M2 are the first and second Zagreb indices, holds for chemical graphs, and Vukičević and Graovac [17] proved that this also holds for trees. In both works is given a distinct counterexample for which this inequality is false in general. Here, we present some classes of graphs with prescribed degrees, that satisfy M1/n ≤ M2/m. Namely every graph G whose degrees of vertices are in the interval [c, c + ⌈ √ c ⌉] for some integer c, satisfies this inequality. In addition, we prove that for any ∆ ≥ 5, there is an infinite family of graphs of maximum degree ∆ such that the inequality is false. Moreover, an alternative and slightly shorter proof for trees is presented, as well as for unicyclic graphs.

with extremal energy should have a small

Let G be a graph on n vertices with r: = ⌊n/2 ⌋ and let λ1 ≥ · · · ≥ λn be adjacency eigenvalues of G. Then the Hückel energy of G, HE(G), is defined as 2

In this paper, it is shown that among connected graphs with maximum clique size ω, the minimum value of the spectral radius of adjacency matrix is attained for a kite graph P Kn−ω,ω, which consists of a complete graph Kω to a vertex of which a path Pn−ω is attached. For any fixed ω, a small interval to which the spectral radii of kites P Km,ω, m ≥ 1, belong is exhibited.