When almost distance-regularity attains distance-regularity

Generally speaking, `almost distance-regular graphs' are graphs which share some, but not necessarily all, regularity properties that characterize distance-regular graphs. In this paper we rst propose four basic di erent (but closely related) concepts of almost distance-regularity. In some cases, they coincide with concepts introduced before by other authors, such as walk-regular graphs and partially distance-regular graphs. Here it is always assumed that the diameter D of the graph attains its maximum possible value allowed by its number d+1 of di erent eigenvalues; that is, D = d, as happens in every distance-regular graph. Our study focuses on nding out when almost distance- regularity leads to distance-regularity. In other words, some `economic' (in the sense of minimizing the number of conditions) old and new characterizations of distance- regularity are discussed. For instance, if A0;A1; : : : ;AD and E0;E1; : : : ;Ed denote, respectively, the distance matrices and the idempotents of the graph; and D and A stand for their respective linear spans, any of the two following `dual' conditions su ce: (a) A0;A1;AD 2 A; (b) E0;E1;Ed 2 D. Moreover, other characterizations based on the preintersection parameters, the average intersection numbers and the recurrence coe cients are obtained. In some cases, our results can be also seen as a generalization of the so-called spectral excess theorem for distance-regular graphs.Postprint (published version

Some applications of the proper and adjacency polynomials in the theory of graph spectra

Given a vertex u\inV of a graph $\Gamma=(V,E)$, the (local) proper polynomials constitute a sequence of orthogonal polynomials, constructed from the so-called $u$-local spectrum of $\Gamma$. These polynomials can be thought of as a generalization, for all graphs, of the distance polynomials for te distance-regular graphs. The (local) adjacency polynomials, which are basically sums of proper polynomials, were recently used to study a new concept of distance-regularity for non-regular graphs, and also to give bounds on some distance-related parameters such as the diameter. Here we develop the subject of these polynomials and gave a survey of some known results involving them. For instance, distance-regular graphs are characterized from their spectra and the number of vertices at ``extremal distance'' from each of their vertices. Afterwards, some new applications of both, the proper and adjacency polynomials, are derived, such as bounds for the radius of $\Gamma$ and the weight $k$-excess of a vertex. Given the integers $k,\mu\ge 0$, let $\Gamma_k^{\mu}(u)$ denote the set of vertices which are at distance at least $k$ from a vertex $u\in V$, and there exist exactly $\mu$ (shortest) $k$-paths from $u$ to each each of such vertices. As a main result, an upper bound for the cardinality of $\Gamma_k^{\mu}(u)$ is derived, showing that $|\Gamma_k^{\mu}(u)|$ decreases at least as $O(\mu^{-2})$, and the cases in which the bound is attained are characterized. When these results are particularized to regular graphs with four distinct eigenvalues, we reobtain a result of Van Dam about $3$-class association schemes, and prove some conjectures of Haemers and Van Dam about the number of vertices at distane three from every vertex of a regular graph with four distinct eigenvalues---setting $k=2$ and $\mu=0$---and, more generally, the number of non-adjacent vertices to every vertex $u\in V$, which have $\mu$ common neighbours with it.Peer Reviewe

Perfect state transfer, graph products and equitable partitions

We describe new constructions of graphs which exhibit perfect state transfer on continuous-time quantum walks. Our constructions are based on variants of the double cones [BCMS09,ANOPRT10,ANOPRT09] and the Cartesian graph products (which includes the n-cube) [CDDEKL05]. Some of our results include: (1) If $G$ is a graph with perfect state transfer at time $t_{G}$, where t_{G}\Spec(G) \subseteq \ZZ\pi, and $H$ is a circulant with odd eigenvalues, their weak product $G \times H$ has perfect state transfer. Also, if $H$ is a regular graph with perfect state transfer at time $t_{H}$ and $G$ is a graph where t_{H}|V_{H}|\Spec(G) \subseteq 2\ZZ\pi, their lexicographic product $G[H]$ has perfect state transfer. (2) The double cone $\overline{K}_{2} + G$ on any connected graph $G$, has perfect state transfer if the weights of the cone edges are proportional to the Perron eigenvector of $G$. This generalizes results for double cone on regular graphs studied in [BCMS09,ANOPRT10,ANOPRT09]. (3) For an infinite family \GG of regular graphs, there is a circulant connection so the graph K_{1}+\GG\circ\GG+K_{1} has perfect state transfer. In contrast, no perfect state transfer exists if a complete bipartite connection is used (even in the presence of weights) [ANOPRT09]. We also describe a generalization of the path collapsing argument [CCDFGS03,CDDEKL05], which reduces questions about perfect state transfer to simpler (weighted) multigraphs, for graphs with equitable distance partitions.Comment: 18 pages, 6 figure

Some applications of linear algebra in spectral graph theory

The application of the theory of matrices and eigenvalues to combinatorics is cer- tainly not new. In the present work the starting point is a theorem that concerns the eigenvalues of partitioned matrices. Interlacing yields information on subgraphs of a graph, and the way such subgraphs are embedded. In particular, one gets bounds on extremal substructures. Applications of this theorem and of some known matrix theorems to matrices associated to graphs lead to new results. For instance, some characterizations of regular partitions, and bounds for some parameters, such as the independence and chromatic numbers, the diameter, the bandwidth, etc. This master thesis is a contribution to the area of algebraic graph theory and the study of some generalizations of regularity in bipartite graphs. In Chapter 1 we recall some basic concepts and results from graph theory and linear algebra. Chapter 2 presents some simple but relevant results on graph spectra concerning eigenvalue interlacing. Most of the previous results that we use were obtained by Haemers in [33]. In that work, the author gives bounds for the size of a maximal (co)clique, the chromatic number, the diameter and the bandwidth in terms of the eigenvalues of the standard adjacency matrix or the Laplacian matrix. He also nds some inequalities and regularity results concerning the structure of graphs. The work initiated by Fiol [26] in this area leads us to Chapter 3. The discussion goes along the same spirit, but in this case eigenvalue interlacing is used for proving results about some weight parameters and weight-regular partitions of a graph. In this master thesis a new observation leads to a greatly simpli ed notation of the results related with weight-partitions. We nd an upper bound for the weight independence number in terms of the minimum degree. Special attention is given to regular bipartite graphs, in fact, in Chapter 4 we contribute with an algebraic characterization of regularity properties in bipartite graphs. Our rst approach to regularity in bipartite graphs comes from the study of its spectrum. We characterize these graphs using eigenvalue interlacing and we pro- vide an improved bound for biregular graphs inspired in Guo's inequality. We prove a condition for existence of a k-dominating set in terms of its Laplacian eigenvalues. In particular, we give an upper bound on the sum of the rst Laplacian eigenvalues of a k-dominating set and generalize a Guo's result for these structures. In terms of predistance polynomials, we give a result that can be seen as the biregular coun- terpart of Ho man's Theorem. Finally, we also provide new characterizations of bipartite graphs inspired in the notion of distance-regularity. In Chapter 5 we describe some ideas to work with a result from linear algebra known as the Rayleigh's principle. We observe that the clue is to make the \right choice" of the eigenvector that is used in Rayleigh's principle. We can use this method 1 to give a spectral characterization of regular and biregular partitions. Applying this technique, we also derive an alternative proof for the upper bound of the independence number obtained by Ho man (Chapter 2, Theorem 1.2). Finally, in Chapter 6 other related new results and some open problems are pre- sented

Eigenvalue interlacing and weight parameters of graphs

Eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics. In particular, it has been successfully used for proving a number of results about the relation between the (adjacency matrix or Laplacian) spectrum of a graph and some of its properties. For instance, some characterizations of regular partitions, and bounds for some parameters, such as the independence and chromatic numbers, the diameter, the bandwidth, etc., have been obtained. For each parameter of a graph involving the cardinality of some vertex sets, we can define its corresponding weight parameter by giving some "weights" (that is, the entries of the positive eigenvector) to the vertices and replacing cardinalities by square norms. The key point is that such weights "regularize" the graph, and hence allow us to define a kind of regular partition, called "pseudo-regular," intended for general graphs. Here we show how to use interlacing for proving results about some weight parameters and pseudo-regular partitions of a graph. For instance, generalizing a well-known result of Lov\'asz, it is shown that the weight Shannon capacity $\Theta^*$ of a connected graph \G, with $n$ vertices and (adjacency matrix) eigenvalues $\lambda_1>\lambda_2\ge\...\ge \lambda_n$, satisfies \Theta\le \Theta^* \le \frac{\|\vecnu\|^2}{1-\frac{\lambda_1}{\lambda_n}} where $\Theta$ is the (standard) Shannon capacity and \vecnu is the positive eigenvector normalized to have smallest entry 1. In the special case of regular graphs, the results obtained have some interesting corollaries, such as an upper bound for some of the multiplicities of the eigenvalues of a distance-regular graph. Finally, some results involving the Laplacian spectrum are derived. spectrum are derived

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