The hierarchical product of graphs

Abstract

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.

Mean first-passage time for random walks on generalized deterministic recursive trees

“…Here we study its generalization as GDRT and we give the exact analytical value for the MFPT for any value of r and any iteration step t. A formal definition of T r,t as the t hierarchical power of the path P r (with respect to the hierarchical product of graphs introduced in [23]), allowed its rigorous topological study in [24]. We adapt spectral techniques, likes those used for the study of the adjacency spectrum of the hierarchical product in [22][23][24], to analyze the Laplacian spectrum of the GDRT.…”

“…), the determination of the exact spectrum of a general graph is a difficult task. However, for some large families of trees, the recursivity of the graph construction helps to find a relationship between the characteristic polynomials at different iteration steps [22][23][24][25][26], which can be used to produce iteratively an analytical expression for the spectra. Here, we apply a similar technique for the generalized deterministic recursive trees (GDRT) introduced in [24] and known as r-adic hypertrees.…”

Self-similar non-clustered planar graphs as models for complex networks

“…Hierarchical modularity also appears in some models based on k-trees or clique-trees, where the graph is constructed by adding at each step one or more vertices and each is connected independently to a certain subgraph [8,10,13]. The introduction of the so-called hierarchical product of graphs in [21] allows a generalization and a rigorous study of some of these models.…”

The Total Irregularity of Some Composite Graphs

“…The hierarchical product of graphs, introduced in [6], is defined as follows. Let G i = (V i , E i ) be graphs with vertex sets V i , i = 1, 2, having a distinguished or root vertex, labeled 0.…”