New results on odd harmonious labeling of graphs

Let G = (V, E) be a graph with p vertices and q edges. A graph G is said to be odd harmonious if there exists an injection f : V (G) → {0, 1, 2, · · · , 2q − 1}such that the induced function f* : E(G) → {1, 3, · · · , 2q − 1} defined by f* (uv) = f(u) + f(v) is a bijection. If f(V (G)) = {0, 1, 2, · · · , q} then f is called strongly odd harmonious labeling and the graph is called strongly odd harmonious graph. In this paper we prove that Spl(Cbn) and Spl(B(m)(n)), slanting ladder SLn, mGn, H-super subdivision of path Pn and cycle Cn, n ≡ 0(mod 4) admit odd harmonious labeling. In addition we observe that all strongly odd harmonious graphs admit mean labeling, odd mean labeling, odd sequential labeling and all odd sequential graphs are odd harmonious and all odd harmonious graphs are even sequential harmonious.Emerging Sources Citation Index (ESCI)MathScinetScopu

A covariant causal set approach to discrete quantum gravity

A covariant causal set (c-causet) is a causal set that is invariant under labeling. Such causets are well-behaved and have a rigid geometry that is determined by a sequence of positive integers called the shell sequence. We first consider the microscopic picture. In this picture, the vertices of a c-causet have integer labels that are unique up to a label isomorphism. This labeling enables us to define a natural metric $d(a,b)$ between time-like separated vertices $a$ and $b$. The time metric $d(a,b)$ results in a natural definition of a geodesic from $a$ to $b$. It turns out that there can be $n\ge 1$ such geodesics. Letting $a$ be the origin (the big bang), we define the curvature $K(b)$ of $b$ to be $n-1$. Assuming that particles tend to move along geodesics, $K(b)$ gives the tendency that vertex $b$ is occupied. In this way, the mass distribution is determined by the geometry of the c-causet. We next consider the macroscopic picture which describes the growth process of c-causets. We propose that this process is governed by a quantum dynamics given by complex amplitudes. At present, these amplitudes are unknown. But if they can be found, they will determine the (approximate) geometry of the c-causet describing our particular universe. As an illustration, we present a simple example of an amplitude process that may have physical relevance. We also give a discrete analogue of Einstein's field equations.Comment: 23 pages, 6 tables; new version corrects some typos in the proof of Theorem 6.

On the category of Euclidean configuration spaces and associated fibrations

We calculate the Lusternik-Schnirelmann category of the k-th ordered configuration spaces F(R^n,k) of R^n and give bounds for the category of the corresponding unordered configuration spaces B(R^n,k) and the sectional category of the fibrations pi^n_k: F(R^n,k) --> B(R^n,k). We show that secat(pi^n_k) can be expressed in terms of subspace category. In many cases, eg, if n is a power of 2, we determine cat(B(R^n,k)) and secat(pi^n_k) precisely.Comment: This is the version published by Geometry & Topology Monographs on 19 March 200

Answering Regular Path Queries on Workflow Provenance

This paper proposes a novel approach for efficiently evaluating regular path queries over provenance graphs of workflows that may include recursion. The approach assumes that an execution g of a workflow G is labeled with query-agnostic reachability labels using an existing technique. At query time, given g, G and a regular path query R, the approach decomposes R into a set of subqueries R1, ..., Rk that are safe for G. For each safe subquery Ri, G is rewritten so that, using the reachability labels of nodes in g, whether or not there is a path which matches Ri between two nodes can be decided in constant time. The results of each safe subquery are then composed, possibly with some small unsafe remainder, to produce an answer to R. The approach results in an algorithm that significantly reduces the number of subqueries k over existing techniques by increasing their size and complexity, and that evaluates each subquery in time bounded by its input and output size. Experimental results demonstrate the benefit of this approach