18 research outputs found
Identifying Codes and Domination in the Product of Graphs
An identifying code in a graph is a dominating set that also has the property that the closed neighborhood of each vertex in the graph has a distinct intersection with the set. The minimum cardinality of an identifying code in a graph is denoted \gid(G). We consider identifying codes of the direct product . In particular, we answer a question of Klav\v{z}ar and show the exact value of \gid(K_n \times K_m). It was recently shown by Gravier, Moncel and Semri that for the Cartesian product of two same-sized cliques \gid(K_n \Box K_n) = \lfloor{\frac{3n}{2}\rfloor}. Letting be any integers, we show that \IDCODE(K_n \cartprod K_m) = \max\{2m-n, m + \lfloor n/2 \rfloor\}. Furthermore, we improve upon the bounds for \IDCODE(G \cartprod K_m) and explore the specific case when is the Cartesian product of multiple cliques. Given two disjoint copies of a graph , denoted and , and a permutation of , the permutation graph is constructed by joining to for all . The graph is said to be a universal fixer if the domination number of is equal to the domination number of for all of . In 1999 it was conjectured that the only universal fixers are the edgeless graphs. We prove the conjecture
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
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
Edgeless graphs are the only universal fixers
summary:Given two disjoint copies of a graph , denoted and , and a permutation of , the graph is constructed by joining to for all . is said to be a universal fixer if the domination number of is equal to the domination number of for all of . In 1999 it was conjectured that the only universal fixers are the edgeless graphs. Since then, a few partial results have been shown. In this paper, we prove the conjecture completely
Worm Colorings
Given a coloring of the vertices, we say subgraph H is monochromatic if every vertex of H is assigned the same color, and rainbow if no pair of vertices of H are assigned the same color. Given a graph G and a graph F, we define an F-WORM coloring of G as a coloring of the vertices of G without a rainbow or monochromatic subgraph H isomorphic to F. We present some results on this concept especially as regards to the existence, complexity, and optimization within certain graph classes. The focus is on the case that F is the path on three vertices