A graph H is said to be light in a family G of graphs if at least one member of G contains a copy of H and there exists an integer λ(H,G) such that each member G of G with a copy of H also has a copy K of H such that degG(v) ≤ λ(H,G) for all v ∈ V(K). In this paper, we study the light graphs in the class of graphs with small average degree, including the plane graphs with some restrictions on g...

Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function f : E(G) −→ {±1,±2, . . . ,±k} is said to be a signed star {k}-dominating function on G if ∑ e∈E(v) f(e) ≥ k for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N(v)}. The signed star {k}-domination number of a graph G is γ{k}SS(G) = min{ ∑ e∈E f(e) | f is a S...

Chang, Y.-W., M.S. Jacobson, CL. Monma and D.B. West, Subtree and substar intersection numbers. Discrete Applied Mathematics 44 (1993) 205-220. We introduce the star number [free number] of a graph, being the minimum t such that G is the intersection graph of unions of t substars [subtrees] of a host tree. We study bounds on these parameters, compare them with interval number, and characterize ...

In a signed graph G, a negative clique is a complete subgraph having negative edges only. In this article, we give characteristic polynomial expressions, and eigenvalues of some signed graphs having negative cliques. This includes signed cycle graph, signed path graph, a complete graph with disjoint negative cliques, and star block graph with negative cliques.

The arrangement graph was proposed as a generalization of the star graph topology. In this paper we investigate the topological properties of the (n, k)-arrangement graph &k. It has been shown that the (n, n 2)arrangement graph An,+2 is isomorphic to the n-alternating group graph AGn In addition, the exact value of average distance of A,,& has been derived.

At Crypto-87 conference, we proposed identity-based key distribution systems for generating a common secret conference key for two or more users. Protocols were shown for three configurations: a ring, a complete graph, and a star. Yacobi has made an impersonation attack on the protocols for the complete graph and star networks. This paper proposes improved identity-based key distribution protoc...

The star graph possess many nice topological properties. Edge fault tolerance is an important issue for a network since the edges in the network may fail sometimes. In this paper, we show that the n-dimensional star graph is (n 3)-edge fault tolerant hamiltonian laceable, (n 3)-edge fault tolerant strongly hamiltonian laceable, and (n 4)edge fault tolerant hyper hamiltonian laceable. All these ...

We solve an open question posed by Akers and Krishnamurthy in 1986 [1, 3] concerning VLSI layout of star graphs. We show that the area of the optimal layout of an N-node star graph, hierarchical cubic network (HCN), or hierarchical folded-hypercube network (HFN) is N2=16 o(N2) under the Thompson model, or under the extended grid model where a node occupies a rectangle of sides that may range fr...

Some problems related to security in communication networks lead to consider a new type of connectivity in graphs, namely the confidential connectivity. In this paper we present a characterization of unbreakable graphs using the notion of weak decomposition and we give some applications of minimal unbreakable graphs. In fact, we showed that a graph G is confidentially connected if and only if i...