The spectral excess theorem for distance-regular graphs states that a regular (connected) graph i... more The spectral excess theorem for distance-regular graphs states that a regular (connected) graph is distance-regular if and only if its spectral-excess equals its average excess. A bipartite graph $\Gamma$ is distance-biregular when it is distance-regular around each vertex and the intersection array only depends on the stable set such a vertex belongs to. In this note we derive a new version of the spectral excess theorem for bipartite distance-biregular graphs.
We give a formal definition of a new product of bipartite digraphs, the Manhat-tan product, and w... more We give a formal definition of a new product of bipartite digraphs, the Manhat-tan product, and we study some of its main properties. It is shown that if all the factors of the above product are (directed) cycles, then the digraph obtained is the Manhattan street network. To this respect, it is proved that many properties of these networks, such as high symmetries and the presence of Hamiltonian cycles, are shared by the Manhattan product of some digraphs. Moreover, we prove that the Manhattan product of two Manhattan streets networks is also a Manhattan street network. Also, some necessary conditions for the Manhattan product of two Cayley digraphs to be again a Cayley digraph are given.
A maximally connected digraph G is said to be super-κ if all its minimum disconnecting sets are t... more A maximally connected digraph G is said to be super-κ if all its minimum disconnecting sets are trivial. Analogously, G is called super-λ if it is maximally arcconnected and all its minimum arc-disconnecting sets are trivial. It is first proved that any bipartite digraph G with diameter D is super-κ if D ≤ 2 − 1, and it is super-λ if D ≤ 2 , where denotes a parameter related to the number of short paths. These results allow us to show that if the order of a bipartite digraph G is big enough then superconnectivity is attained. For instance, if G is d-regular and has diameter D = 3 and ≥ 1, then G is super-λ if n > 4d; and if D = 4 and ≥ 2, then G is super-κ if n > 4d 2. In these cases the results are proved to be best possible. Similar results are given for bipartite (undirected) graphs. (For a graph it turns out that = (g − 2)/2, where g stands for the girth.
The extraconnectivity κ(n) of a simple connected graph G is a kind of conditional connectivity wh... more The extraconnectivity κ(n) of a simple connected graph G is a kind of conditional connectivity which is the minimum cardinality of a set of vertices, if any, whose deletion disconnects G in such a way that every remaining component has more than n vertices. The usual connectivity and superconnectivity of G correspond to κ(0) and κ(1) respectively. This paper gives sufficient conditions, relating the diameter D, the girth g, and the minimum degree δ of a graph, to assure maximum extraconnectivity. For instance, if D ≤ g − n + 2(δ − 3), being n ≥ 2δ + 4 and g ≥ n + 5, then the value of κ(n) is (n+1)δ−2n, which is optimal. The corresponding edge version of this result, to assure maximum edge-extraconnectivity λ(n), is also discussed.
Let Γ be a graph on n vertices, adjacency matrix A, and distinct eigenvalues λ > λ 1 > λ 2 > • • ... more Let Γ be a graph on n vertices, adjacency matrix A, and distinct eigenvalues λ > λ 1 > λ 2 > • • • > λ d. For every k = 0, 1,. .. , d − 1, the k-alternating polynomial P k is defined to be the polynomial of degree k and norm P k ∞ = max 1≤l≤d {|P k (λ l)|} = 1 that attains maximum value at λ. These polynomials, which may be thought of as the discrete version of the Chebychev ones, were recently used by the authors to bound the diameter D(Γ) of Γ in terms of its eigenvalues. Namely, it was shown that P k (λ) > ν 2 − 1 ⇒ D(Γ) ≤ k, where ν is the (positive) eigenvector associated to λ with minimum component 1. In this work we improve upon this result by assuming that some extra information about the structure of Γ is known. To this end, we introduce the so-called τ-adjacency polynomial Q τ. For each 0 ≤ τ ≤ d, the polynomial Q τ is defined to be the polynomial of degree τ and norm Q τ A = max 1≤i≤n { Q τ (A)e i } = 1 that attains maximum value at λ. Then it is shown that P k (λ) > ν 2 /Q 2 τ (λ) − 1 ⇒ D(Γ) ≤ k + 2τ. Some applications of the above results, together with new bounds for generalized diameters, are also presented.
We comment on the paper "Extremal Cayley digraphs of finite Abelian groups" [Intercon. ... more We comment on the paper "Extremal Cayley digraphs of finite Abelian groups" [Intercon. Networks 12 (2011), no. 1-2, 125--135]. In particular, we give some counterexamples to the results presented there, and provide a correct result for degree two.
The concept of (local) pseudo-distance-regularity, recently introduced, is a natural generalizati... more The concept of (local) pseudo-distance-regularity, recently introduced, is a natural generalization of distance-regularity, intended for not necessarily regular graphs. We study here some properties of locally pseudo-distance-regular graphs and give some new characterization of such structures. As a consequence, some new characterizations of distance-regular graphs are also derived.
Let D6(G) be the Cayley colored ugraph of a finite group G generated by A. The arc-colored line d... more Let D6(G) be the Cayley colored ugraph of a finite group G generated by A. The arc-colored line digraph of a Cayley colored digraph ie obtained by appropriately coloring the arcs of its line digraph. In this paper it is shown that the group of automorphisms of D6 (G) that act as permutations on the color classes is isomorphic to the gemidirect product of G aird a particular subgroup of AutG. Neceeeary and sufficient conditions for the arc-colored une digraph of a Cayley colored digraph ?lso to be a Cayley colored digraph are then derived.
Eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics. ... more 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 resul...
In this paper we study the spectral properties of a family of trees characterized by two main fea... more In this paper we study the spectral properties of a family of trees characterized by two main features: they are spanning subgraphs of the hypercube, and their vertices bear a high degree of (connectedness) hierarchy. Such structures are here called binary hypertrees and they can be recursively defined as the so-called hierarchical product of several complete graphs on two vertices.
This letter deals with the relationship between the total number of k-walks in a graph, and the s... more This letter deals with the relationship between the total number of k-walks in a graph, and the sum of the k-th powers of its vertex degrees. In particular, it is shown that the sum of all k-walks is upper bounded by the sum of the k-th powers of the degrees.
We derive some Moore-like bounds for multipartite digraphs, which extend those of bipartite digra... more We derive some Moore-like bounds for multipartite digraphs, which extend those of bipartite digraphs, under the assumption that every vertex of a given partite set is adjacent to the same number δ of vertices in each of the other independent sets. We determine when a Moore multipartite digraph is weakly distance-regular. Within this framework, some necessary conditions for the existence of a Moore r-partite digraph with interpartite outdegree δ> 1 and diameter k = 2m are obtained. In the case δ = 1, which corresponds to almost Moore digraphs, a necessary condition in terms of the permutation cycle structure is derived. Additionally, we present some constructions of dense multipartite digraphs of diameter two that are vertex-transitive.
We study the relationship between two key concepts in the theory of digraphs, those of quotient d... more We study the relationship between two key concepts in the theory of digraphs, those of quotient digraphs and voltage digraphs. These techniques contract or expand a given digraph in order to study its characteristics,or to obtain more involved structures. As an application, we relate the spectrum of a digraph Γ, called a voltage digraph or base, with the spectrum of its lifted digraph Γα. We prove that all the eigenvalues of Γ (including multiplicities) are, in addition, eigenvalues of Γα. This study is carried out by introducing several reduced matrix representations of Γα. As an example of our techniques, we study some basic properties of the Alegre digraph and its base.
We present some related families of orthogonal polynomials of a discrete variable and survey some... more We present some related families of orthogonal polynomials of a discrete variable and survey some of their applications in the study of (distance-regular) graphs and (completely regular) codes. One of the main peculiarities of such orthogonal systems is their non-standard normalization condition, requiring that the square norm of each polynomial must equal its value at a given point of the mesh. For instance, when they are defined from the spectrum of a graph, one of these families is the system of the pre-distance polinomials which, in the case of distance-regular graphs, turns out to be the sequence of distance polinomials. The applications range from (quasi-spectral) characterizations of distance-regular graphs, walk-regular graphs, local distance-regularity and completely regular codes, to some results on representation theory.
Eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics. ... more 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 s~aow how to use interlacing for proving resu...
Se describen algunas aplicaciones de la teor¶‡a de matrices a diversos temas pertenecientes alde ... more Se describen algunas aplicaciones de la teor¶‡a de matrices a diversos temas pertenecientes alde la matematica discreta. En particular, se van a con- siderar las siguientesde investigacion: (a) La teor¶‡a de matrices enteras, con las formas normales de Hermite y de Smith, es utilizada en el estudio de la congruencia de vectores enteros, con su generalizacion del teorema chino del resto y su relacion con grupos abelianos. Sus posibles aplicaciones van desde el dise~no de ciertas redes delocal, pasando por la teor¶‡a de mosaicos y equidescomposicide flguras planas, hasta el desarrollo de metodos de encrip- tado en criptograf¶‡a; (b) La teor¶‡a espectral de matrices se utiliza ampliamente en diversos temas de la teor¶‡a de grafos. En particular, se ha utilizado reciente- mente para el estudio y caracterizacion de grafos distancia-regulares, esquemas de asociacion, y sus aplicaciones en teor¶‡a de codigos completamente regulares; (c) La teor¶‡a de matrices circulantes tiene multiples ap...
Eigenvalue interlacing is a versatile technique for deriving results in algebraic com-binatorics.... more Eigenvalue interlacing is a versatile technique for deriving results in algebraic com-binatorics. 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 interlac-ing for provi...
Abelian Cayley digraphs can be constructed by using a generalization to $\mathbb{Z}^n$ of the con... more Abelian Cayley digraphs can be constructed by using a generalization to $\mathbb{Z}^n$ of the concept of congruence in $\mathbb{Z}$. Here we use this approach to present a family of such digraphs, which, for every fixed value of the degree, have asymptotically large number of vertices as the diameter increases. Up to now, the best known large dense results were all non-constructive.
We comment on the paper “Extremal Cayley digraphs of finite Abelian groups” [Intercon. Networks 1... more We comment on the paper “Extremal Cayley digraphs of finite Abelian groups” [Intercon. Networks 12 (2011), no. 1-2, 125–135]. In particular, we give some counterexamples to the results presented there, and provide a correct result for degree two.
The spectral excess theorem for distance-regular graphs states that a regular (connected) graph i... more The spectral excess theorem for distance-regular graphs states that a regular (connected) graph is distance-regular if and only if its spectral-excess equals its average excess. A bipartite graph $\Gamma$ is distance-biregular when it is distance-regular around each vertex and the intersection array only depends on the stable set such a vertex belongs to. In this note we derive a new version of the spectral excess theorem for bipartite distance-biregular graphs.
We give a formal definition of a new product of bipartite digraphs, the Manhat-tan product, and w... more We give a formal definition of a new product of bipartite digraphs, the Manhat-tan product, and we study some of its main properties. It is shown that if all the factors of the above product are (directed) cycles, then the digraph obtained is the Manhattan street network. To this respect, it is proved that many properties of these networks, such as high symmetries and the presence of Hamiltonian cycles, are shared by the Manhattan product of some digraphs. Moreover, we prove that the Manhattan product of two Manhattan streets networks is also a Manhattan street network. Also, some necessary conditions for the Manhattan product of two Cayley digraphs to be again a Cayley digraph are given.
A maximally connected digraph G is said to be super-κ if all its minimum disconnecting sets are t... more A maximally connected digraph G is said to be super-κ if all its minimum disconnecting sets are trivial. Analogously, G is called super-λ if it is maximally arcconnected and all its minimum arc-disconnecting sets are trivial. It is first proved that any bipartite digraph G with diameter D is super-κ if D ≤ 2 − 1, and it is super-λ if D ≤ 2 , where denotes a parameter related to the number of short paths. These results allow us to show that if the order of a bipartite digraph G is big enough then superconnectivity is attained. For instance, if G is d-regular and has diameter D = 3 and ≥ 1, then G is super-λ if n > 4d; and if D = 4 and ≥ 2, then G is super-κ if n > 4d 2. In these cases the results are proved to be best possible. Similar results are given for bipartite (undirected) graphs. (For a graph it turns out that = (g − 2)/2, where g stands for the girth.
The extraconnectivity κ(n) of a simple connected graph G is a kind of conditional connectivity wh... more The extraconnectivity κ(n) of a simple connected graph G is a kind of conditional connectivity which is the minimum cardinality of a set of vertices, if any, whose deletion disconnects G in such a way that every remaining component has more than n vertices. The usual connectivity and superconnectivity of G correspond to κ(0) and κ(1) respectively. This paper gives sufficient conditions, relating the diameter D, the girth g, and the minimum degree δ of a graph, to assure maximum extraconnectivity. For instance, if D ≤ g − n + 2(δ − 3), being n ≥ 2δ + 4 and g ≥ n + 5, then the value of κ(n) is (n+1)δ−2n, which is optimal. The corresponding edge version of this result, to assure maximum edge-extraconnectivity λ(n), is also discussed.
Let Γ be a graph on n vertices, adjacency matrix A, and distinct eigenvalues λ > λ 1 > λ 2 > • • ... more Let Γ be a graph on n vertices, adjacency matrix A, and distinct eigenvalues λ > λ 1 > λ 2 > • • • > λ d. For every k = 0, 1,. .. , d − 1, the k-alternating polynomial P k is defined to be the polynomial of degree k and norm P k ∞ = max 1≤l≤d {|P k (λ l)|} = 1 that attains maximum value at λ. These polynomials, which may be thought of as the discrete version of the Chebychev ones, were recently used by the authors to bound the diameter D(Γ) of Γ in terms of its eigenvalues. Namely, it was shown that P k (λ) > ν 2 − 1 ⇒ D(Γ) ≤ k, where ν is the (positive) eigenvector associated to λ with minimum component 1. In this work we improve upon this result by assuming that some extra information about the structure of Γ is known. To this end, we introduce the so-called τ-adjacency polynomial Q τ. For each 0 ≤ τ ≤ d, the polynomial Q τ is defined to be the polynomial of degree τ and norm Q τ A = max 1≤i≤n { Q τ (A)e i } = 1 that attains maximum value at λ. Then it is shown that P k (λ) > ν 2 /Q 2 τ (λ) − 1 ⇒ D(Γ) ≤ k + 2τ. Some applications of the above results, together with new bounds for generalized diameters, are also presented.
We comment on the paper "Extremal Cayley digraphs of finite Abelian groups" [Intercon. ... more We comment on the paper "Extremal Cayley digraphs of finite Abelian groups" [Intercon. Networks 12 (2011), no. 1-2, 125--135]. In particular, we give some counterexamples to the results presented there, and provide a correct result for degree two.
The concept of (local) pseudo-distance-regularity, recently introduced, is a natural generalizati... more The concept of (local) pseudo-distance-regularity, recently introduced, is a natural generalization of distance-regularity, intended for not necessarily regular graphs. We study here some properties of locally pseudo-distance-regular graphs and give some new characterization of such structures. As a consequence, some new characterizations of distance-regular graphs are also derived.
Let D6(G) be the Cayley colored ugraph of a finite group G generated by A. The arc-colored line d... more Let D6(G) be the Cayley colored ugraph of a finite group G generated by A. The arc-colored line digraph of a Cayley colored digraph ie obtained by appropriately coloring the arcs of its line digraph. In this paper it is shown that the group of automorphisms of D6 (G) that act as permutations on the color classes is isomorphic to the gemidirect product of G aird a particular subgroup of AutG. Neceeeary and sufficient conditions for the arc-colored une digraph of a Cayley colored digraph ?lso to be a Cayley colored digraph are then derived.
Eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics. ... more 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 resul...
In this paper we study the spectral properties of a family of trees characterized by two main fea... more In this paper we study the spectral properties of a family of trees characterized by two main features: they are spanning subgraphs of the hypercube, and their vertices bear a high degree of (connectedness) hierarchy. Such structures are here called binary hypertrees and they can be recursively defined as the so-called hierarchical product of several complete graphs on two vertices.
This letter deals with the relationship between the total number of k-walks in a graph, and the s... more This letter deals with the relationship between the total number of k-walks in a graph, and the sum of the k-th powers of its vertex degrees. In particular, it is shown that the sum of all k-walks is upper bounded by the sum of the k-th powers of the degrees.
We derive some Moore-like bounds for multipartite digraphs, which extend those of bipartite digra... more We derive some Moore-like bounds for multipartite digraphs, which extend those of bipartite digraphs, under the assumption that every vertex of a given partite set is adjacent to the same number δ of vertices in each of the other independent sets. We determine when a Moore multipartite digraph is weakly distance-regular. Within this framework, some necessary conditions for the existence of a Moore r-partite digraph with interpartite outdegree δ> 1 and diameter k = 2m are obtained. In the case δ = 1, which corresponds to almost Moore digraphs, a necessary condition in terms of the permutation cycle structure is derived. Additionally, we present some constructions of dense multipartite digraphs of diameter two that are vertex-transitive.
We study the relationship between two key concepts in the theory of digraphs, those of quotient d... more We study the relationship between two key concepts in the theory of digraphs, those of quotient digraphs and voltage digraphs. These techniques contract or expand a given digraph in order to study its characteristics,or to obtain more involved structures. As an application, we relate the spectrum of a digraph Γ, called a voltage digraph or base, with the spectrum of its lifted digraph Γα. We prove that all the eigenvalues of Γ (including multiplicities) are, in addition, eigenvalues of Γα. This study is carried out by introducing several reduced matrix representations of Γα. As an example of our techniques, we study some basic properties of the Alegre digraph and its base.
We present some related families of orthogonal polynomials of a discrete variable and survey some... more We present some related families of orthogonal polynomials of a discrete variable and survey some of their applications in the study of (distance-regular) graphs and (completely regular) codes. One of the main peculiarities of such orthogonal systems is their non-standard normalization condition, requiring that the square norm of each polynomial must equal its value at a given point of the mesh. For instance, when they are defined from the spectrum of a graph, one of these families is the system of the pre-distance polinomials which, in the case of distance-regular graphs, turns out to be the sequence of distance polinomials. The applications range from (quasi-spectral) characterizations of distance-regular graphs, walk-regular graphs, local distance-regularity and completely regular codes, to some results on representation theory.
Eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics. ... more 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 s~aow how to use interlacing for proving resu...
Se describen algunas aplicaciones de la teor¶‡a de matrices a diversos temas pertenecientes alde ... more Se describen algunas aplicaciones de la teor¶‡a de matrices a diversos temas pertenecientes alde la matematica discreta. En particular, se van a con- siderar las siguientesde investigacion: (a) La teor¶‡a de matrices enteras, con las formas normales de Hermite y de Smith, es utilizada en el estudio de la congruencia de vectores enteros, con su generalizacion del teorema chino del resto y su relacion con grupos abelianos. Sus posibles aplicaciones van desde el dise~no de ciertas redes delocal, pasando por la teor¶‡a de mosaicos y equidescomposicide flguras planas, hasta el desarrollo de metodos de encrip- tado en criptograf¶‡a; (b) La teor¶‡a espectral de matrices se utiliza ampliamente en diversos temas de la teor¶‡a de grafos. En particular, se ha utilizado reciente- mente para el estudio y caracterizacion de grafos distancia-regulares, esquemas de asociacion, y sus aplicaciones en teor¶‡a de codigos completamente regulares; (c) La teor¶‡a de matrices circulantes tiene multiples ap...
Eigenvalue interlacing is a versatile technique for deriving results in algebraic com-binatorics.... more Eigenvalue interlacing is a versatile technique for deriving results in algebraic com-binatorics. 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 interlac-ing for provi...
Abelian Cayley digraphs can be constructed by using a generalization to $\mathbb{Z}^n$ of the con... more Abelian Cayley digraphs can be constructed by using a generalization to $\mathbb{Z}^n$ of the concept of congruence in $\mathbb{Z}$. Here we use this approach to present a family of such digraphs, which, for every fixed value of the degree, have asymptotically large number of vertices as the diameter increases. Up to now, the best known large dense results were all non-constructive.
We comment on the paper “Extremal Cayley digraphs of finite Abelian groups” [Intercon. Networks 1... more We comment on the paper “Extremal Cayley digraphs of finite Abelian groups” [Intercon. Networks 12 (2011), no. 1-2, 125–135]. In particular, we give some counterexamples to the results presented there, and provide a correct result for degree two.
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Papers by Miguel Fiol