A novel approach to complex problems has been previously applied to graph classification and the graph equivalence problem. Here we apply it to the NP complete problem of finding the largest perfect clique within a graph G. PACS numbers: 89.75.Hc, 89.90.+n, 46.70.-p, 95.75.Pq [email protected] [email protected] [email protected] 2

Let G be the set of finite graphs whose vertices belong to some fixed countable set, and let ≡ be an equivalence relation on G. By the strengthening of ≡ we mean an equivalence relation ≡s such that G ≡s H , where G,H ∈ G, if for every F ∈ G, G ∪ F ≡ H ∪ F . The most important case that we study in this paper concerns equivalence relations defined by graph properties. We write G ≡ H , where Φ i...

The cycle and co-cycle matroids of a bond graph are defined using chain group matroids derived,from the cycle and co-cycle vector spaces qf a bond graph. The relationship between these structures is investigated and various results are proved. A precise equivalence is dejined for bond graphs. Duality theory is seen to be very clear in the context of bond graph matroids.

The local complementation rule is applied for continuous-variable (CV) graph states in the paper, which is an elementary graph transformation rule and successive application of which generates the orbit of any graph states. The corresponding local Gaussian transformations of local complementation for four-mode unweighted graph states were found, which do not mirror the form of the local Cliffor...

The category D of finite directed graphs is cartesian closed, hence it has a product and exponential objects. For a fixed K, let K be the class of all directed graphs of the formK, preordered by the existence of homomorphisms, and quotiented by homomorphic equivalence. It has loong been known that K, is always boolean lattice. In this paper we prove that for any complete graph Kn with n ≥ 3, K ...

In this paper, we determine two families R and G of subsets of a hyperring R and sufficient conditions such that two geometric spaces (R,R) and (R,G) are strongly transitive. Moreover, we prove that the relations Γ and α are strongly regular equivalence relations on a hyperfield or a hyperring such that (R,+) has an identity element.