Greedy Trees, Subtrees and Antichains

The greedy tree $\mathcal{G}(D)$ and the $\mathcal{M}$-tree $\mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which $\mathcal{G}(D)$ or $\mathcal{M}(D)$ is extremal. Many known results, for example on the Wiener index, the number of subtrees, the number of independent subsets and the number of matchings follow as corollaries, as do some new results on invariants such as the number of rooted spanning forests, the incidence energy and the solvability. We also extend our results on trees with fixed degree sequence $D$ to the set of trees whose degree sequence is majorised by a given sequence $D$, which also has a number of applications.

A note on extremal trees with degree conditions

This paper presents a general recursive formula to estimate the number of labeled graphs as well as details to evaluate the formula for the following graph properties: number of edges (graph density), degree sequence, degree distribution, classification mixing, and degree mixing, i.e., the formula estimates the number of labeled graphs that have given values for graph properties. The proposed approach can be extended to additional graph properties (e.g., number of triangles) as well as properties of bipartite graphs. For special settings in which formulas exist from previous research, simulation studies demonstrate the validity of the proposed approach. In addition, we demonstrate how our approach can be used to quantify the level of variability in values of a graph property in the subset of graphs that hold a specified value of a different graph property (or properties) constant.

A channel scheduling problem for a given graph G(V,E) is to select a subset E′ of edges of E and assign channel to each one in E′ under the restriction that all edges of E′ are interference free. We introduce the notion of two sides approximation for the channel scheduling problem. A pair of parameters (f,g) controls the accuracy of approximation in this paper. A (f,g)-approximation satisfies \(\sum_{e\in E'}W(e)\ge{\sum_{e\in Opt}W(e)\over f}\) and \(\sum_{e\in E-E'}W(e)\le g\sum_{e\in E- Opt^*}W(e)\), where Opt * is the set of edges assigned channels in an optimal solution, and W(e) is the weight of edge e. An f-approximation satisfies \(\sum_{e\in E'}W(e)\ge{\sum_{e\in Opt}W(e)\over f}\). We show that a simple greedy algorithm can obtain an (O(1), O(1))-approximation for the single channel scheduling problem. In many cases, the greedy algorithms gives much more accurate result than the worst ratio. Furthermore, we develop an \(|E|^{O({1\over \epsilon})}\) time (1 − ε, O(1))-approximation algorithm for the single channel scheduling problem. We also show that a simple greedy algorithm can obtain an O(1)-approximation for the multi-channel scheduling problem which satisfies \(\sum_{e\in E'}W(e)\ge{\sum_{e\in Opt}W(e)\over \Omega(1)}\). We also develop a \(|E|^{O({d\over \epsilon})}\) time (1 − ε)-approximation algorithm the multi-channel scheduling problem, where d is the number of channels. This improves the existing approximation scheme for multi-channel scheduling problem with \(|E|^{O({d\over \epsilon^2})}\) time by Cheng et al. We also develop a polynomial time constant factor greedy approximation algorithm for the multi-channel scheduling that allows variate radii of interference among those nodes.

Difference graphs

In this note, we discuss the method explained in the recent paper [P. Manuel, I. Rajasingh, B. Rajan, R. Sundara Rajan, A New Approach To Compute Wiener Index, Journal of Computational and Theoretical Nanoscience 10, (2013) 1515-1521.] for computing the Wiener index of special chemical graphs. The method is actually already well-known and equivalent to the 'cut method' introduced in 1995 by Klavzar, Gutman and Mohar, and used in multiple papers for computing various distance based graph invariants.

Nanostructures are objects whose sizes vary between microscopic and molecular. The most significant of these new elements are carbon nanotubes. These elements have extraordinary microelectronic properties and many other exclusive physiognomies. Recently, researchers have given attention to the mathematical properties of these materials. The aim and objective of this research article is to investigate the most important molecular descriptors namely Wiener, edge-Wiener, vertex-edge-Wiener, vertex-Szeged, edge-Szeged, edge-vertex-Szeged, total-Szeged, PI, Schultz, Gutman, Mostar, edge-Mostar, and total-Mostar indices of three-layered single-walled titania nanosheets. By computing these topological indices, material science researchers can have a better understanding of structural and physical properties of titania nanosheets, thereby synthesizing more easily new variants of titania nanosheets with more amenable physicochemical properties. The cut method turned out to be extremely handy when dealing with distance-based graph invariants which are in turn among the central concepts of chemical graph theory. In this method, we use the Djokovi-Winkler relation to find the suitable edge cuts to leave the graph into exactly two components. Based on the graph theoretical measures of the components, we obtain the desired topological indices by mathematical computations. In this paper, distance-based indices for three-layered single-walled titania nanosheets were investigated and given the exact expressions for various dimensions of three-layered singlewalled titania nanosheets. These indices may be useful in synthesizing new variants of titania nanosheets and the computed topological indices play an important role in studies of Quantitative structure-activity relationship (QSAR) and Quantitative structure-property relationship (QSPR). In this paper, we have obtained the closed expressions of several distance-based topological indices of three-layered single-walled titania nanosheet TNS3 [m, n] molecular graph for the cases m > n and m < n. The graphical validations for the computed indices are done and we observe that the Wiener types, Schultz and Gutman indices perform in a similar way whereas PI and Mostar type indices perform in the same way.

Inequalities provide a way to study topological indices relatively. There are two major classes of topological indices: degree-based and distance-based indices. In this paper we provide a relative study of these classes and derive inequalities between degree-based indices such as Randić connectivity, GA, ABC, and harmonic indices and distance-based indices such as eccentric connectivity, connective eccentric, augmented eccentric connectivity, Wiener, and third ABC indices.

Relay stations can be deployed in a wireless network to extend its coverage and improve its capacity. In this paper, we study a scheduling problem in OFDMA-based wireless relay networks with consideration for multi-user diversity, channel diversity and spatial reuse. First, we present a Mixed Integer Linear Programming (MILP) formulation to provide optimum solutions. It has been shown by previous research that performance of a wireless scheduling algorithm is usually related to the interference degree delta, which is the maximum number of links that interfere with a common link but do not interfere with each other. Therefore, we then show that the interference degree delta is at most 4 for any 2-hop relay network and 14 for any general h-hop (h >; 1) relay network. Furthermore, we present a simple greedy algorithm for the scheduling problem and show it has an approximation ratio of 1/(1+δ), which leads to an approximation ratio of 1/5 for the 2-hop case and 1/15} for the general case. In addition, we present three heuristic algorithms, namely, the weighted degree greedy algorithm, the Maximum Weighted Independent Set (MWIS) algorithm and the Linear Programming (LP) rounding algorithm, to solve the scheduling problem. Extensive simulation results have showed that the LP rounding algorithm performs best and always provides close-to-optimum solutions. The performance of the simple greedy algorithm is comparable to that of the other algorithms.

In this paper, we investigate the structures of extremal trees which have the minimal number of subtrees in the set of all trees with a given degree sequence. In particular, the extremal trees must be caterpillar and but in general not unique. Moreover, all extremal trees with a given degree sequence $${\pi = (d_1, \ldots, d_{5}, 1, \ldots, 1)}$$ ? = ( d 1 , ? , d 5 , 1 , ? , 1 ) have been characterized.

Hamiltonian properties of 3-connected {claw,hourglass}-free graphs

Replicating of objects at multi nodes has been considered a promising technique for decreasing access cost and network traffic for a large distributed systems like Data Grids. In this paper we propose Optimal Placement of Replicas (OPR) algorithm with the objective of minimizing the total replication cost including communication and storage cost, given that a maximum of k replicas are allowed in a tree data grid with n nodes and height h. OPR has time complexity of O(nhk), which is the lowest time complexity compared with the other algorithms with the same objective function in the past literature. In addition, two greedy algorithms, namely, Greedy Algorithm (GA) and Simple Greedy Algorithm (SGA) are proposed for near optimal placement of k replicas with the lower time complexity compared with that of OPR. We evaluate the proposed algorithms through an extensive simulation experiments in terms of access cost, network traffic and locality access to objects.

Influence maximization problem is to find a small set of influential nodes in a social network, whose activation of information propagation can be maximized using one of the propagation models such as independent cascade model. This paper proposes a new method to solve this problem based on discretized linear automaton algorithm and simple greedy algorithm. Every allowable node in the social network is regarded as an action and the goal of the learning automaton is to select an optimal subset of actions. To speed up the convergence, firstly, the problem space is reduced by excluding the nodes lacking influence; then, the original learning process is divided into two parts: the sampling process and the learning process. Besides, a new scheme for determining the best parameters is presented, which is more applicable to practical problems. The obtained results show that the new proposed algorithm is efficient in real-life social networks. The results are close to the ones obtained by the greedy algorithm in terms of accuracy, and the new method greatly improves the speed of selecting the optimal subset.

Efficient operation of wireless networks and switches requires using simple (and in some cases distributed) scheduling algorithms. In general, simple greedy algorithms (known as Greedy Maximal Scheduling - GMS) are guaranteed to achieve only a fraction of the maximum possible throughput (e.g., 50% throughput in switches). However, it was recently shown that in networks in which the Local Pooling conditions are satisfied, GMS achieves 100% throughput. Moreover, in networks in which the &#963;- Local Pooling conditions hold, GMS achieves &#963;% throughput. In this paper, we focus on identifying the specific network topologies that satisfy these conditions. In particular, we provide the first characterization of all the network graphs in which Local Pooling holds under primary interference constraints (in these networks GMS achieves 100% throughput). This leads to a polynomial time algorithm for identifying Local Pooling-satisfying graphs. Moreover, by using similar graph theoretical methods, we show that in all bipartite graphs (i.e., input-queued switches) of size up to 7×n, GMS is guaranteed to achieve 66% throughput, thereby improving upon the previously known 50% lower bound. Finally, we study the performance of GMS in interference graphs and show that in certain specific topologies its performance could be very bad. Overall, the paper demonstrates that using graph theoretical techniques can significantly contribute to our understanding of greedy scheduling algorithms.

The well‐known lower bound on the independence number of a graph due to Caro (Technical Report, Tel‐Aviv University, 1979) and Wei (Technical Memorandum, TM 81 ‐ 11217 ‐ 9, Bell Laboratories, 1981) can be established as a performance guarantee of two natural and simple greedy algorithms or of a simple randomized algorithm. We study possible generalizations and improvements of these approaches using vertex weights and discuss conditions on so‐called potential functions pG: V(G)→ℕ0 defined on the vertex set of a graph G for which suitably modified versions of the greedy algorithms applied to G yield independent sets I with . We provide examples of such potentials, which lead to bounds improving the bound due to Caro and Wei. Furthermore, suitably adapting the randomized algorithm we give a short proof of Thiele's lower bound on the independence number of a hypergraph (T. Thiele, J Graph Theory 30 (1999), 213–221).