We provide in Appendix 1 a list of additional situations in which an algebraic property of L C ( E ) is analogous to a topological property of C ∗ ( E ), and for which the necessary and sufficient graph-theoretic property of E is identical in each case.
This condition turns out to be sufficient for graphs with at most three maximal cliques.
We present new necessary and sufficient conditions for a graph to be controllable relative to its adjacency matrix or to its signless Laplacian without having to evaluate any eigenspaces, which is the criterion usually employed.
But this kind of information is not sufficient to comprehend the graph.
Therefore, we consider a more generic strategy to estimate pairwise distance, which is sufficient to perform guided graph exploration.
According to definition 6 and definition 7, dmin(G) > 0 is a necessary condition instead of a sufficient condition that communication graph G is connected.
In this paper, we give necessary and sufficient conditions for the graph H u, n to be connected and a forest.
T1 consists of potential neighbors of T0 which are stored in B1 "by mistake", i.e. belongb to B1 but not to T0. B1 and T1 are sufficient to represent the graph provided that the only queried k-mers are those which are potential neighbors of k-mers of T0.
In this paper, we get a sufficient condition on topological graphs so that the associated C∗-algebras are simple and purely infinite.
Implementations of some of the most complex analysis algorithms known are presented, showing that a language based on graphs and graph transformations is sufficient for performing flow analysis.
By utilizing the multiple Lyapunov functions method and algebraic graph theory, a sufficient condition for consensus tracking is established.
