For a graph $G$ let $gamma (G)$ be its domination number. We define a graph G to be (i) a hypo-efficient domination graph (or a hypo-$mathcal{ED}$ graph) if $G$ has no efficient dominating set (EDS) but every graph formed by removing a single vertex from $G$ has at least one EDS, and (ii) a hypo-unique domination graph (a hypo-$mathcal{UD}$ graph) if $G$ has at least two minimum dominating sets...

For a fixed positive integer n, let Wn be the permutation matrix corresponding to the permutation ( 1 2 · · · n− 1 n 2 3 · · · n 1 ) . In this article, it is shown that a symmetric matrix with rational entries is circulant if, and only if, it lies in the subalgebra of Q[x]/〈x−1〉 generated by Wn+W −1 n . On the basis of this, the singularity of graphs on n-vertices is characterized algebraically...

In this work, we consider the problem of K-sparse graph signal recovery on perturbed circulant graphs, simulating network clusters within a large network, as an extension of the framework of sampling and reconstructing signals with a finite rate of innovation (FRI) in the classical domain to the graph domain. In light of the fact that the GFT-basis on a circulant graph G = (V,E) is, up to a per...

A circulant is a Cayley graph over a cyclic group. A well-covered graph is a graph in which all maximal stable sets are of the same size α = α(G), or in other words, they are all maximum. A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. It is not difficult to show that a circulant G is a CIS graph if and only if G and its complement G are both well-co...

We investigate a certain condition for isomorphism between circulant graphs (known as the Ad am property) in a stronger form by referring to isospectrality rather than to isomorphism of graphs. We describe a wide class of graphs for which the Ad am conjecture holds. We apply these results to establish an asymptotic formula for the number of non-isomorphic circulant graphs and connected circulan...

A graph is well-covered if every independent set can be extended to a maximum independent set. We show that it is co-NP-complete to determine whether an arbitrary graph is well-covered, even when restricted to the family of circulant graphs. Despite the intractability of characterizing the complete set of well-covered circulant graphs, we apply the theory of independence polynomials to show tha...

We propose geometrical methods for constructing square 01-matrices with the same number n of units in every row and column, and such that any two rows of the matrix have at most one unit in the same position. In terms of Design Theory, such a matrix is an incidence matrix of a symmetric configuration. Also, it gives rise to an n-regular bipartite graphs without 4-cycles, which can be used for c...

For the interconnection network topology, it is usually represented by a graph. When a network is used, processors and/or links faults may happen. Thus, it is meaningful to consider faulty networks. We consider k-regular graphs in this paper. We define a k-regular hamiltonian and hamiltonian connected graph G is super fault-tolerant hamiltonian if G remains hamiltonian after removing at most k ...

The clique graph K(G) of a graph G is the intersection graph of all its (maximal) cliques, and G is said to be clique divergent if the order of its n-th iterated clique graph Kn(G) tends to infinity with n. In general, deciding whether a graph is clique divergent is not known to be computable. We characterize the dynamical behavior under the clique operator of circulant graphs of the form Cn(a,...