Complete Split Graphs Research Articles

Spectral Characterization of Families of Split Graphs

An upper bound for the sum of the squares of the entries of the principal eigenvector corresponding to a vertex subset inducing a k-regular subgraph is introduced and applied to the determination of an upper bound on the order of such induced subgraphs. Furthermore, for some connected graphs we establish a lower bound for the sum of squares of the entries of the principal eigenvector corresponding to the vertices of an independent set. Moreover, a spectral characterization of families of split graphs, involving its index and the entries of the principal eigenvector corresponding to the vertices of the maximum independent set is given. In particular, the complete split graph case is highlighted.

Proof of the first part of the conjecture of Aouchiche and Hansen about the Randić index

Let G(k,n) be the set of connected simple n-vertex graphs with minimum vertex degree k. The Randić index R(G) of a graph G is defined by: R(G)=∑uv∈E(G)1d(u)d(v), where d(u) is the degree of vertex u and the summation extends over all edges uv of G. In this paper we prove for k≤n2 the conjecture of Aouchiche and Hansen about the graphs in G(k,n) for which the Randić index attains its minimum value. We show that the extremal graphs are complete split graphs Kk,n−k∗, which have only two degrees, i.e. degree k and degree n−1, and the number of vertices of degree k is n−k, while the number of vertices of degree n−1 is k. At the end we generalize our results to graphs with prescribed maximum degree q.

Infinite families of [formula omitted]-integral graphs

In this article, we characterize all signless Laplacian integral graphs (here called Q -integral graphs) in the following classes: complete split graphs, multiple complete split-like graphs, extended complete split-like graphs, multiple extended split-like graphs. All these graphs were defined by Hansen et al. [P. Hansen, H. Melot, D, Stevanović, Integral complete split graphs, Univ. Beograd, Publ. Elektrotehn. Fak. Ser. Mat. 13 (2002) 89-95]. Also, we characterize all Q -integral graphs of the form K n 1 ∨ ( K n 2 ∪ K n 3 ) , where K n i , i = 1 , 2 , 3 , is a complete graph on n i vertices. These characterizations allow us to exhibit many infinite families of Q -integral graphs.

Integral complete split graphs

We give characterizations of integral graphs in the family of complete split graphs and a few related families of graphs.

The minimum broadcast time problem for several processor networks

Broadcasting is the information dissemination process in a communication network. A subset of processors V 0 ⊂ V called originators knows an unique message which has to be transferred by calls between adjacent processors. Each call requires one time unit and each processor can participate in at most one call per time unit. The problem is to find a schedule such that the time needed to inform all processors is less than or equal to a deadline k ≠ N . We present NP-completeness results for this problem restricted to several communication networks (bipartite planar graphs, grid graphs, complete grid graphs, split graphs and chordal graphs) with constant deadline k = 2 or one originator V 0 = { υ }.

Absolute retracts of split graphs

It is proved that a split graph is an absolute retract of split graphs if and only if a partition of its vertex set into a stable set and a complete set is unique or it is a complete split graph. Three equivalent conditions for a split graph to be an absolute retract of the class of all graphs are given. It is finally shown that a reflexive split graph G is an absolute retract of reflexive split graphs if and only if G has no retract isomorphic to some J n , n⩾3. Here J n is the reflexive graph with vertex set { x 1, x 2,…, x n , y 1, y 2,…, y n } in which the vertices x 1, x 2,…, x n are mutually adjacent and the vertex y i is adjacent to x 1, x 2,…, x i−1 , x i+1 ,…, x n .

Edge domination in complete partite graphs

An edge dominating set in a graph G is a set of edges D such that every edge not in D is adjacent to an edge of D. An edge domatic partition of a graph G=( V, E) is a collection of pairwise-disjoint edge dominating sets of G whose union is E. The maximum size of an edge domatic partition of G is called the edge domatic number. In this paper, we study the edge domatic number of the complete partite graphs and give the answers for balanced complete partite graphs and complete split graphs.

ON THE INTRACTABILITY OF TASK ALLOCATION IN DISTRIBUTED SYSTEMS

The fast progress of large integration technology has made distributed computing economically attractive for many computer applications. Task allocation is one of the most important and challenging problems in distributed computing systems that has received considerable attention in recent years. Several researchers have introduced different heuristics to solve this problem with the assumption that it is computationally intractable. These researchers have been referring to an early work by Stone and some private communication as their sources of the problem’s intractability. However, neither Stone’s paper nor any other related work in the literature provided a formal proof of intractability. Due to the importance of the task allocation problem, we believe that a formal proof of its complexity must be provided. In this paper we provide a proof that the problem of task allocation is intractable even in the restricted case when there are only two values of the the communication cost: zero or one. We introduce a new model to represent the problem of allocating tasks on heterogeneous distributed systems using split graphs. This model allows us to relate the task allocation problem with the problem of weighted clique partitioning in complete split graphs. We provide a two-step reduction method to prove the intractability of task allocation in the restricted case mentioned above. In the first step, we prove that the problem of weighted clique partitioning in complete split graph is NP-complete using a transformation from the problem of partition into triangles. In the second step, we show that the task allocation problem is NP-complete using a transformation from the weighted clique partitioning problem.