44 research outputs found
Spectral properties of the hierarchical product of graphs
The hierarchical product of two graphs represents a natural way to build a
larger graph out of two smaller graphs with less regular and therefore more
heterogeneous structure than the Cartesian product. Here we study the
eigenvalue spectrum of the adjacency matrix of the hierarchical product of two
graphs. Introducing a coupling parameter describing the relative contribution
of each of the two smaller graphs, we perform an asymptotic analysis for the
full spectrum of eigenvalues of the adjacency matrix of the hierarchical
product. Specifically, we derive the exact limit points for each eigenvalue in
the limits of small and large coupling, as well as the leading-order relaxation
to these values in terms of the eigenvalues and eigenvectors of the two smaller
graphs. Given its central roll in the structural and dynamical properties of
networks, we study in detail the Perron-Frobenius, or largest, eigenvalue.
Finally, as an example application we use our theory to predict the epidemic
threshold of the Susceptible-Infected-Susceptible model on a hierarchical
product of two graphs
The hierarchical product of graphs
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
(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
Spectral properties of the hierarchical product of graphs
The hierarchical product of two graphs represents a natural way to build a larger graph out of two smaller graphs with less regular and therefore more heterogeneous structure than the Cartesian product. Here we study the eigenvalue spectrum of the adjacency matrix of the hierarchical product of two graphs. Introducing a coupling parameter describing the relative contribution of each of the two smaller graphs, we perform an asymptotic analysis for the full spectrum of eigenvalues of the adjacency matrix of the hierarchical product. Specifically, we derive the exact limit points for each eigenvalue in the limits of small and large coupling, as well as the leading-order relaxation to these values in terms of the eigenvalues and eigenvectors of the two smaller graphs. Given its central roll in the structural and dynamical properties of networks, we study in detail the Perron-Frobenius, or largest, eigenvalue. Finally, as an example application we use our theory to predict the epidemic threshold of the susceptible-infected-susceptible model on a hierarchical product of two graphs
On the Hierarchical Product of Graphs and the Generalized Binomial Tree
In this paper we follow the study of the hierarchical product of graphs, an operation recently introduced in the context of networks. A well-known example of such a product is the binomial tree which is the (hierarchical) power of the complete graph on two vertices. An
appealing property of this structure is that all the eigenvalues are distinct. Here we show how to obtain a graph with this property by applying the hierarchical product. In particular, we propose a generalization of the binomial tree and some of its main properties are studied
The Wiener, Eccentric Connectivity and Zagreb Indices of the Hierarchical Product of Graphs
Let G1 = (V1, E1) and G2 = (V2, E2) be two graphs having a distinguished or root vertex, labeled 0. The hierarchical product G2 ⊓ G1
of G2 and G1 is a graph with vertex set V2 × V1. Two vertices y2y1 and x2x1 are adjacent if and only if y1x1 ∈ E1 and y2 = x2; or y2x2 ∈ E2 and y1 = x1 = 0. In this paper, the Wiener, eccentric connectivity and Zagreb indices of this new operation of graphs are computed. As an application, these topological indices for a class of alkanes are computed. ACM Computing Classification System (1998): G.2.2, G.2.3.The research of this paper is partially supported by the University of Kashan under grant no 159020/12
Revan weighted PI index on some product of graphs
In chemical graph thoery, PI index is an additive topological index which has been used to measure the characteristics of chemical compounds. In this paper we introduce the weighted version of PI index of graph called the Revan Weighted PI index and we have obtained it for the hierarchical product of graphs, cartesian product, subdivision and join of two graphs. Also we have derived this index for some molecular graphs.Publisher's Versio
On the metric dimension and fractional metric dimension for hierarchical product of graphs
A set of vertices {\em resolves} a graph if every vertex of is
uniquely determined by its vector of distances to the vertices in . The {\em
metric dimension} for , denoted by , is the minimum cardinality of
a resolving set of . In order to study the metric dimension for the
hierarchical product of two rooted graphs
and , we first introduce a new parameter, the {\em
rooted metric dimension} \rdim(G_1^{u_1}) for a rooted graph . If
is not a path with an end-vertex , we show that
\dim(G_2^{u_2}\sqcap G_1^{u_1})=|V(G_2)|\cdot\rdim(G_1^{u_1}), where
is the order of . If is a path with an end-vertex ,
we obtain some tight inequalities for .
Finally, we show that similar results hold for the fractional metric dimension.Comment: 11 page
Metric Dimension of Amalgamation of Graphs
A set of vertices resolves a graph if every vertex is uniquely
determined by its vector of distances to the vertices in . The metric
dimension of is the minimum cardinality of a resolving set of .
Let be a finite collection of graphs and each
has a fixed vertex or a fixed edge called a terminal
vertex or edge, respectively. The \emph{vertex-amalgamation} of , denoted by , is formed by taking all
the 's and identifying their terminal vertices. Similarly, the
\emph{edge-amalgamation} of , denoted by
, is formed by taking all the 's and identifying
their terminal edges.
Here we study the metric dimensions of vertex-amalgamation and
edge-amalgamation for finite collection of arbitrary graphs. We give lower and
upper bounds for the dimensions, show that the bounds are tight, and construct
infinitely many graphs for each possible value between the bounds.Comment: 9 pages, 2 figures, Seventh Czech-Slovak International Symposium on
Graph Theory, Combinatorics, Algorithms and Applications (CSGT2013), revised
version 21 December 201