### Bounds for the Hückel energy 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 ap ..."

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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.

### inequality ∗

, 2009

"... 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 inequal ..."

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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.

### HE(G) =

, 2009

"... 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 ..."

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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

### THE MINIMUM SPECTRAL RADIUS OF GRAPHS WITH A GIVEN CLIQUE NUMBER

, 2008

"... 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 interv ..."

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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.

### Bounds on the Q-spread of a graph

"... The spread s(M) of an n × n complex matrix M is s(M) = maxij|λi − λj|, where the maximum is taken over all pairs of eigenvalues of M, λi,1 ≤ i ≤ n, [9] and [11]. Based on this concept, Gregory et al. [7] determined some bounds for the spread of the adjacency matrix A(G) of a simple graph G and made ..."

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The spread s(M) of an n × n complex matrix M is s(M) = maxij|λi − λj|, where the maximum is taken over all pairs of eigenvalues of M, λi,1 ≤ i ≤ n, [9] and [11]. Based on this concept, Gregory et al. [7] determined some bounds for the spread of the adjacency matrix A(G) of a simple graph G and made a conjecture regarding the graph on n vertices yielding the maximum value of the spread of the corresponding adjacency matrix. The signless Laplacian matrix of a graph G, Q(G) = D(G)+A(G), where D(G) is the diagonal matrix of degrees of G and A(G) is its adjacency matrix, has been recently studied, [4], [5]. The main goal of this paper is to determine some bounds on s(Q(G)). We prove that, for any graph on n ≥ 5 vertices, 2 ≤ s(Q(G)) ≤ 2n − 4, and we characterize the equality cases in both bounds. Further, we prove that for any connected graph G with n ≥ 5 vertices, s(Q(G)) < 2n − 4. We conjecture that, for n ≥ 5, sQ(G) ≤ √ 4n 2 − 20n + 33 and that, in this case, the upper bound is attained if, and only if, G is a certain pathcomplete graph. Key words: spectrum, signless Laplacian matrix, spread, path complete graph

### Methods and Applications

, 2008

"... Les textes publiés dans la série des rapports de recherche HEC n’engagent que la responsabilité de leurs auteurs. La publication de ces rapports de recherche bénéficie d’une subvention du Fonds québécois de la recherche sur la nature et les technologies. Variable Neighborhood Search: ..."

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Les textes publiés dans la série des rapports de recherche HEC n’engagent que la responsabilité de leurs auteurs. La publication de ces rapports de recherche bénéficie d’une subvention du Fonds québécois de la recherche sur la nature et les technologies. Variable Neighborhood Search:

### BICLIQUE COMPLETION PROBLEM: MODELS AND ALGORITHMS

, 2009

"... 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 ..."

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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

### HYBRID META-HEURISTICS FOR UNIFORM PARALLEL MACHINE TO MINIMIZE TOTAL WEIGHTED COMPLETION TIME

- EVALUATION AND OPTIMIZATION OF INNOVATIVE PRODUCTION SYSTEMS OF GOODS AND SERVICES
, 2010

"... In this paper, we consider the problem of scheduling n independent jobs on m uniform parallel machines such that total weighted completion time is minimized. We present two meta-heuristics and two hybrid meta-heuristics to solve this problem. Based on a set of instances, a comparative study has be ..."

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In this paper, we consider the problem of scheduling n independent jobs on m uniform parallel machines such that total weighted completion time is minimized. We present two meta-heuristics and two hybrid meta-heuristics to solve this problem. Based on a set of instances, a comparative study has been realized in order to evaluate these approaches.