Spectral characterization of families of split graphs

An upper bound for the sum of the squares of the entries of the principal eigenvector corresponding to a vertex subset inducing a k-regular subgraph is introduced and applied to the determination of an upper bound on the order of such induced subgraphs. Furthermore, for some connected graphs we establish a lower bound for the sum of squares of the entries of the principal eigenvector corresponding to the vertices of an independent set. Moreover, a spectral characterization of families of split graphs, involving its index and the entries of the principal eigenvector corresponding to the vertices of the maximum independent set is given. In particular, the complete split graph case is highlighted

Resultados espectrais relacionados com a estrutura dos grafos

Doutoramento em MatemáticaNesta tese são estabelecidas novas propriedades espectrais de grafos com estruturas específicas, como sejam os grafos separados em cliques e independentes e grafos duplamente separados em independentes, ou ainda grafos com conjuntos (κ,τ)-regulares. Alguns invariantes dos grafos separados em cliques e independentes são estudados, tendo como objectivo limitar o maior valor próprio do espectro Laplaciano sem sinal. A técnica do valor próprio é aplicada para obter alguns majorantes e minorantes do índice do espectro Laplaciano sem sinal dos grafos separados em cliques e independentes bem como sobre o índice dos grafos duplamente separados em independentes. São fornecidos alguns resultados computacionais de modo a obter uma melhor percepção da qualidade desses mesmos extremos. Estudamos igualmente os grafos com um conjunto (κ,τ)-regular que induz uma estrela complementar para um valor próprio não-principal $. Além disso, é mostrado que$=κ-τ. Usando uma abordagem baseada nos grafos estrela complementares construímos, em alguns casos, os respectivos grafos maximais. Uma caracterização dos grafos separados em cliques e independentes que envolve o índice e as entradas do vector principal é apresentada tal como um majorante do número da estabilidade dum grafo conexo.In this thesis new spectral properties of graphs with a specific structure (as split graphs, nested split and double split graphs as well as graphs with (κ,τ)-regular sets) are deduced. Some invariants of nested split graphs are studied in order to bound the largest eigenvalue of signless Laplacian spectra. The eigenvalue technique is applied to obtain some lower and upper bounds on the index of signless Laplacian spectra of nested split graphs as well as on the index of double nested graphs. Computational results are provided in order to gain a better insight of quality of these bounds. The graphs having a (κ,τ)-regular set which induces a star complement for a non-main eigenvalue $are studied. Furthermore, it is shown that$= κ-τ. By the star complement technique, in some cases, maximal graphs with desired properties are constructed. A spectral characterization of families of split graphs involving its index and the entries of the principal eigenvector is given as well as an upper bound on the stability number of a connected graph

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Spectral threshold dominance, Brouwer's conjecture and maximality of Laplacian energy

The Laplacian energy of a graph is the sum of the distances of the eigenvalues of the Laplacian matrix of the graph to the graph's average degree. The maximum Laplacian energy over all graphs on $n$ nodes and $m$ edges is conjectured to be attained for threshold graphs. We prove the conjecture to hold for graphs with the property that for each $k$ there is a threshold graph on the same number of nodes and edges whose sum of the $k$ largest Laplacian eigenvalues exceeds that of the $k$ largest Laplacian eigenvalues of the graph. We call such graphs spectrally threshold dominated. These graphs include split graphs and cographs and spectral threshold dominance is preserved by disjoint unions and taking complements. We conjecture that all graphs are spectrally threshold dominated. This conjecture turns out to be equivalent to Brouwer's conjecture concerning a bound on the sum of the $k$ largest Laplacian eigenvalues

Efficient algorithms for deciding the type of growth of products of integer matrices

For a given finite set $\Sigma$ of matrices with nonnegative integer entries we study the growth of $$\max_t(\Sigma) = \max\{\|A_{1}... A_{t}\|: A_i \in \Sigma\}.$$ We show how to determine in polynomial time whether the growth with $t$ is bounded, polynomial, or exponential, and we characterize precisely all possible behaviors.Comment: 20 pages, 4 figures, submitted to LA