Abstract

A vertex magic total labeling of a graph G = (V, E) is a bijection f: V ≼ E → {1, 2,…, |V| + |E|} such that for every vertex w, the sum is a constant. It is well known that all complete graphs Kn admit a vertex magic total labeling. In this paper we present a new proof of this theorem using the concepts of twin factorization and magic square.

Abstract

Let G = (V, E) be a graph and let f: V → {0, 1} be a mapping from the set of vertices to {0, 1} and for each edge (u, v) ∊ E assign the label |f(u) — f(v)|. If the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1 and the number of edges labled with 0 and the number of edges labeled with 1 differ by at most 1, then f is called a cordial labeling. In this paper, we present two families of planar graphs, Pln and Plm,n defined shortly, that admit a cordial labeling. We also show that the class Plm,n admits a total product cordial labeling under certain conditions.

Abstract

Graph theoretic concepts have been applied to the dynamic cross-correlation data obtained from MD simulation of adenine riboswitch, in absence and presence of adenine. Our hybrid approach combines community detection algorithms that support edge weights and cliques. The effect of variations in the values of nearest neighbors (NN) and correlation coefficient threshold (T) in the community detection algorithm have been applied to identify and filter out coincidental correlations between rogue nodes. Our results have identified the correlations within the structural regions of the molecule, which provide strong clues regarding the functionality and stability of the molecule in absence and presence of adenine. Our results show that a prior application of our algorithm applied to the simulation data of biomolecules in automated fashion can provide strong leads for hypothesis formulation and subsequent hypothesis-driven manual investigation.

Abstract

This paper focuses on anti-magic labellings of planar graphs of type (a, b, c) introduced by Lih (Util. Math. 24 (1983), 165–197). We show that a certain class of planar graphs defined using the complete graphs and a class of planar graphs defined using the complete bipartite graphs are (1, 1, 0) and (1, 1, 1)-anti-magic under certain mild conditions.

Abstract

We address the problem of message authentication in communication networks which are resource constrained or are performance bound. Recent research has focused on development of symmetric key protocols for authentication in such networks. In these protocols, the sender generates a pool of keys -used to sign the messages, and distributes a different subset of keys -used to verify the signatures, to each user. However, in these protocols, users can collude to combine their keys and impersonate the sender by generating the sender signatures. In this work, we describe a family of collusion resistant symmetric key distribution protocols for authentication which address the problem of collusion. We show that the collusion resistance achieved using our protocols is practical (and hence, sufficient) for networks whose communication diameter is known or is within fixed bounds. Furthermore, we show that some existing protocols in literature are members of our family of protocols.

Abstract

We consider the problem of probabilistic reliable communication (PRC) over synchronous networks modeled as directed graphs in the presence of a Byzantine adversary when players’ knowledge of the network topology is not complete. We show that possibility of PRC is extremely sensitive to the changes in players’ knowledge of the topology. This is in complete contrast with earlier known results on the possibility of perfectly reliable communication over undirected graphs where the case of each player knowing only its neighbours gives the same result as the case where players have complete knowledge of the network. Specifically, in either case,(2t+ 1)-vertex connectivity is necessary and sufficient, where t is the number of nodes that can be corrupted by the adversary [DDWY93, SKR+05]. We introduce a novel model for quantifying players’ knowledge of network topology, denoted by T K. Given a directed graph G, influenced by a Byzantine adversary that can corrupt up to any t players, we give a necessary and sufficient condition for possibility of PRC over G for any arbitrary topology knowledge T K. It follows from our main characterization theorem that knowledge of up to d=⌊ n− 2t

Abstract

This paper considers the problem of unicasting in wireless ad hoc networks. Unicasting is the problem of finding a route between a source and a destination and forwarding the message from the source to the destination. In theory, models that have been used oversimplify the problem of route discovery in ad hoc networks. The achievement of this paper is threefold. First we use a more general model in which nodes can have different transmission and interference ranges and we present a new routing algorithm for wireless ad hoc networks that has several nice features. We then combine our algorithm with that of known greedy algorithms to arrive at an average case efficient routing algorithm in the situation that GPS information is available. Finally we show how to schedule unicast traffic between a set of source-destination pairs by providing a proper vertex coloring of the nodes in the wireless ad hoc network. Our coloring algorithm achieves a O(Δ)–coloring that is locally distinct within the 2-hop neighborhood of any node.