A recursive construction of the regular exceptional graphs with least eigenvalue -2

In spectral graph theory a graph with least eigenvalue 2 is exceptional if it is connected, has least eigenvalue greater than or equal to 2, and it is not a generalized line graph. A ðk; tÞ-regular set S of a graph is a vertex subset, inducing a k-regular subgraph such that every vertex not in S has t neighbors in S. We present a recursive construction of all regular exceptional graphs as successive extensions by regular sets.

Main eigenvalues and (κ, τ)-regular sets

A (κ, τ)-regular set is a subset of the vertices of a graph G, inducing a κ-regular subgraph such that every vertex not in the subset has τ neighbors in it. A main eigenvalue of the adjacency matrix A of a graph G has an eigenvector not orthogonal to the all-one vector j. For graphs with a (κ, τ)-regular set a necessary and sufficient condition for an eigenvalue be non-main is deduced and the main eigenvalues are characterized. These results are applied to the construction of infinite families of bidegreed graphs with two main eigenvalues and the same spectral radius (index) and some relations with strongly regular graphs are obtained. Finally, the determination of (κ, τ)-regular sets is analyzed. © 2009 Elsevier Inc. All rights reserved.

Resultados espectrais relacionados com a estrutura dos grafos

Nesta 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.

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.

Necessary and sufficient conditions for a Hamiltonian graph

A graph is singular if the zero eigenvalue is in the spectrum of its 0-1 adjacency matrix A. If an eigenvector belonging to the zero eigenspace of A has no zero entries, then the singular graph is said to be a core graph. A ( k,t)-regular set is a subset of the vertices inducing a k -regular subgraph such that every vertex not in the subset has t neighbours in it. We consider the case when k=t which relates to the eigenvalue zero under certain conditions. We show that if a regular graph has a ( k,k )-regular set, then it is a core graph. By considering the walk matrix we develop an algorithm to extract ( k,k )-regular sets and formulate a necessary and sufficient condition for a graph to be Hamiltonian.

Spectra of graphs obtained by a generalization of the join graph operation

Taking a Fiedler’s result on the spectrum of a matrix formed from two symmetric matrices as a motivation, a more general result is deduced and applied to the determination of adjacency and Laplacian spectra of graphs obtained by a generalized join graph operation on families of graphs (regular in the case of adjacency spectra and arbitrary in the case of Laplacian spectra). Some additional consequences are explored, namely regarding the largest eigenvalue and algebraic connectivity.

Relations between (κ, τ)-regular sets and star complements

Let G be a finite graph with an eigenvalue μ of multiplicity m. A set X of m vertices in G is called a star set for μ in G if μ is not an eigenvalue of the star complement G\X which is the subgraph of G induced by vertices not in X. A vertex subset of a graph is (k ,t)-regular if it induces a k -regular subgraph and every vertex not in the subset has t neighbors in it. We investigate the graphs having a (k,t)-regular set which induces a star complement for some eigenvalue. A survey of known results is provided and new properties for these graphs are deduced. Several particular graphs where these properties stand out are presented as examples.

On the Faria's inequality for the Laplacian and signless Laplacian spectra: a unified approach

Let p(G)p(G) and q(G)q(G) be the number of pendant vertices and quasi-pendant vertices of a simple undirected graph G, respectively. Let m_L±(G)(1) be the multiplicity of 1 as eigenvalue of a matrix which can be either the Laplacian or the signless Laplacian of a graph G. A result due to I. Faria states that mL±(G)(1) is bounded below by p(G)−q(G). Let r(G) be the number of internal vertices of G. If r(G)=q(G), following a unified approach we prove that mL±(G)(1)=p(G)−q(G). If r(G)>q(G) then we determine the equality mL±(G)(1)=p(G)−q(G)+mN±(1), where mN±(1) denotes the multiplicity of 1 as eigenvalue of a matrix N±. This matrix is obtained from either the Laplacian or signless Laplacian matrix of the subgraph induced by the internal vertices which are non-quasi-pendant vertices. Furthermore, conditions for 1 to be an eigenvalue of a principal submatrix are deduced and applied to some families of graphs.

Efficient domination through eigenvalues

The paper begins with a new characterization of (k,τ)(k,τ)-regular sets. Then, using this result as well as the theory of star complements, we derive a simplex-like algorithm for determining whether or not a graph contains a (0,τ)(0,τ)-regular set. When τ=1τ=1, this algorithm can be applied to solve the efficient dominating set problem which is known to be NP-complete. If −1−1 is not an eigenvalue of the adjacency matrix of the graph, this particular algorithm runs in polynomial time. However, although it does not work in polynomial time in general, we report on its successful application to a vast set of randomly generated graphs.

Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Consider two graphs G and H. Let H^k[G] be the lexicographic product of H^k and G, where H^k is the lexicographic product of the graph H by itself k times. In this paper, we determine the spectrum of H^k[G]H and H^k when G and H are regular and the Laplacian spectrum of H^k[G] and H^k for G and H arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, 10^100 ) of vertices is determined. The paper finishes with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.