A new operation on graphs is introduced and some of its properties are studied. We call it hierarchical product, because of the strong (connectedness) hierarchy of the vertices in the resulting graphs. In fact, the obtained graphs turn out to be subgraphs of the cartesian product of the corresponding factors. Some well-known properties of the cartesian product, such as a reduced mean distance and diameter, simple routing algorithms and some optimal communication protocols are inherited by the hierarchical product. We also address the study of some algebraic properties of the hierarchical product of two or more graphs. In particular, the spectrum of the binary hypertree T m (which is the hierarchical product of several copies of the complete graph on two vertices) is fully characterized; turning out to be an interesting example of graph with all its eigenvalues distinct. Finally, some natural generalizations of the hierarchic product are proposed.

Distance-regular graphs are a key concept in Algebraic Combinatorics\ud and have given rise to several generalizations, such as association\ud schemes. Motivated by spectral and other algebraic characterizations\ud of distance-regular graphs, we study ‘almost distanceregular\ud graphs’. We use this name informally for graphs that share\ud some regularity properties that are related to distance in the graph.\ud For example, a known characterization of a distance-regular graph\ud is the invariance of the number of walks of given length between\ud vertices at a given distance, while a graph is called walk-regular if\ud the number of closed walks of given length rooted at any given vertex\ud is a constant. One of the concepts studied here is a generalization\ud of both distance-regularity and walk-regularity called m-walkregularity.\ud Another studied concept is that of m-partial distanceregularity\ud or, informally, distance-regularity up to distance m. Using\ud eigenvalues of graphs and the predistance polynomials, we discuss\ud and relate these and other concepts of almost distance-regularity,\ud such as their common generalization of ( ,m)-walk-regularity. We\ud introduce the concepts of punctual distance-regularity and punctual\ud walk-regularity as a fundament upon which almost distanceregular\ud graphs are built. We provide examples that are mostly\ud taken from the Foster census, a collection of symmetric cubic\ud graphs. Two problems are posed that are related to the question of\ud when almost distance-regular becomes whole distance-regular. We\ud also give several characterizations of punctually distance-regular\ud graphs that are generalizations of the spectral excess theorem.Postprint (published version

A generalization of both the hierarchical product and the Cartesian product of graphs is introduced and some of its properties are studied. We call it the generalized hierarchical product. In fact, the obtained graphs turn out to be subgraphs of the Cartesian product of the corresponding factors. Thus, some well-known properties of this product, such as a good connectivity, reduced mean distance, radius and diameter, simple routing algorithms and some optimal communication protocols, are inherited by the generalized hierarchical product. Besides some of these properties, in this paper we study the spectrum, the existence of Hamiltonian cycles, the chromatic number and index, and the connectivity of the generalized hierarchical product.

We present a method to derive the complete spectrum of the lift Γ α of a base digraph Γ, with voltage assignments on a (finite) group G. The method is based on assigning to Γ a quotient-like matrix whose entries are elements of the group algebra C[G], which fully represents Γ α . This allows us to derive the eigenvectors and eigenvalues of the lift in terms of those of the base digraph and the irreducible characters of G. Thus, our main theorem generalize some previous results of Lováz and Babai concerning the spectra of Cayley digraphs.

Considering a connected graph $G$ with diameter $D$, we say that it is $k$-walk-regular, for a given integer $k$ $(0\leq k \leq D)$, if the number of walks of length $\ell$ between any pair of vertices only depends on the distance between them, provided that this distance does not exceed $k$. Thus, for $k=0$, this definition coincides with that of walk-regular graph, where the number of cycles of length $\ell$ rooted at a given vertex is a constant through all the graph. In the other extreme, for $k=D$, we get one of the possible definitions for a graph to be distance-regular. In this paper we show some algebraic characterizations of $k$-walk-regularity, which are based on the so-called local spectrum and predistance polynomials of $G$.