A graph G of order n and size m is edge-magic if there is a bijection l : V (G) ∪E(G) → [n+m] such that all sums l(a) + l(b) + l(ab), ab ∈ E(G), are the same. We present new lower and upper bounds on M(n), the maximum size of an edge-magic graph of order n, being the first to show an upper bound of the form M(n) ≤ (1 − ǫ) ( n 2 ) . Concrete estimates for ǫ can be obtained by knowing s(k, n), th...

Let G = (V,E) be a graph of order n. A distance magic labeling of G is a bijection l : V → {1, . . . , n} for which there exists a positive integer k such that ∑ x∈N(v) l(x) = k for all v ∈ V , where N(v) is the neighborhood of v. We introduce a natural subclass of distance magic graphs. For this class we show that it is closed for the direct product with regular graphs and closed as a second f...

Given a graph G $G$ , decomposition of is partition its edges. A ( d h ) $(d,h)$ -decomposable if edge set can be partitioned into $d$ -degenerate and with maximum degree at most $h$ . For ≤ 4 $d\le 4$ we are interested in the minimum integer ${h}_{d}$ such that every planar $(d,{h}_{d})$ -decomposable. It was known 3 ${h}_{3}\le 2 8 ${h}_{2}\le 8$ 1 = ∞ ${h}_{1}=\infty $ This paper proves ${h}...

This paper makes two contributions to the computational geometry of decomposable graphs, aimed primarily at facilitating statistical inference about such graphs where they arise as assumed conditional independence structures in stochastic models. The first of these provides sufficient conditions under which it is possible to completely connect two disconnected cliques of vertices, or perform th...

Mycielski introduced a new graph transformation μ(G) for graph G, which is called the Mycielskian of G. A graph G is super connected or simply super-κ (resp. super edge connected or super-λ), if every minimum vertex cut (resp. minimum edge cut) isolates a vertex of G. In this paper, we show that for a connected graph G with |V (G)| ≥ 2, μ(G) is super-κ if and only if δ(G) < 2κ(G), and μ(G) is s...

This paper deals with two types of graph labelings namely, the super (a, d)-edge antimagic total labeling and super (a, d)-vertex antimagic total labeling on the Harary graph C n. We also construct the super edge-antimagic and super vertex-antimagic total labelings for a disjoint union of k identical copies of the Harary graph.

This thesis consists of four papers studying structure learning and Bayesian inference in probabilistic graphical models for both undirected and directed acyclic graphs (DAGs). Paper A presents a novel algorithm, called the Christmas tree algorithm (CTA), that incrementally construct junction trees for decomposable graphs by adding one node at a time to the underlying graph. We prove that CTA w...

A graph is called magic (supermagic) if it admits a labelling of the edges by pairwise different (and consecutive) positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In the paper we prove that any balanced bipartite graph with minimum degree greater than |V (G)|/4 ≥ 2 is magic. A similar result is presented for supermag...

A vertex-magic total labeling of a graph G(V; E) is a one-to-one map from E ∪V onto the integers {1; 2; : : : ; |E|+ |V |} such that (x) + ∑ (xy); where the sum is over all vertices y adjacent to x, is a constant, independent of the choice of vertex x. In this paper we examine the existence of vertex-magic total labelings of trees and forests. The situation is quite di9erent from the conjecture...