Forwarding and optical indices of a graph

Motivated by wavelength-assignment problems for all-to-all traffic in optical networks, we study graph parameters related to sets of paths connecting all pairs of vertices. We consider sets of both undirected and directed paths, under minimisation criteria known as edge congestion and wavelength count; this gives rise to four parameters of a graph G: its edge forwarding index π(G), arc forwarding index , undirected optical index , and directed optical index .In the paper we address two long-standing open problems: whether the equality holds for all graphs, and whether indices π(G) and are hard to compute. For the first problem, we give an example of a family of planar graphs {G_k} such that . For the second problem, we show that determining either π(G) or is NP-hard.     

Inducing regulation of any digraphs

For a given structure D (digraph, multidigraph, or pseudodigraph) and an integer r large enough, a smallest inducing r-regularization of D is constructed. This regularization is an r-regular superstructure of the smallest possible order with bounded arc multiplicity, and containing D as an induced substructure. The sharp upper bound on the number, ρ, of necessary new vertices among such superstructures for n-vertex general digraphs D is determined, ρ being called the inducing regulation number of D. For being the maximum among semi-degrees in D, simple n-vertex digraphs D with largest possible ρ are characterized if either or (where the case is not a trivial subcase of ).     

Two classes of edge domination in graphs

Let (, resp.) be the number of (local) signed edge domination of a graph G [B. Xu, On signed edge domination numbers of graphs, Discrete Math. 239 (2001) 179-189]. In this paper, we prove mainly that and hold for any graph G of order n(n?4), and pose several open problems and conjectures.     

Covering line graphs with equivalence relations

An equivalence graph is a disjoint union of cliques, and the equivalence number of a graph G is the minimum number of equivalence subgraphs needed to cover the edges of G. We consider the equivalence number of a line graph, giving improved upper and lower bounds: . This disproves a recent conjecture that is at most three for triangle-free G; indeed it can be arbitrarily large.To bound we bound the closely related invariant σ(G), which is the minimum number of orientations of G such that for any two edges e,f incident to some vertex v, both e and f are oriented out of v in some orientation. When G is triangle-free, . We prove that even when G is triangle-free, it is NP-complete to decide whether or not σ(G)≤3.     

Complete solution to a problem on the maximal energy of unicyclic bipartite graphs

The energy of a simple graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Denote by C_n the cycle, and the unicyclic graph obtained by connecting a vertex of C₆ with a leaf of P_n-6. Caporossi et al. conjectured that the unicyclic graph with maximal energy is for n=8,12,14 and n≥16. In Hou et al. (2002) [Y. Hou, I. Gutman, C. Woo, Unicyclic graphs with maximal energy, Linear Algebra Appl. 356 (2002) 27-36], the authors proved that is maximal within the class of the unicyclic bipartite n-vertex graphs differing from C_n. And they also claimed that the energies of C_n and is quasi-order incomparable and left this as an open problem. In this paper, by utilizing the Coulson integral formula and some knowledge of real analysis, especially by employing certain combinatorial techniques, we show that the energy of is greater than that of C_n for n=8,12,14 and n≥16, which completely solves this open problem and partially solves the above conjecture.