Sum Of Labels Research Articles

Exclusive Sum Labeling of Graphs: A Survey

All sum graphs are disconnected. In order for a connected graph to bear a sum labeling, the graph is considered in conjunction with a number of isolated vertices, the labels of which complete the sum labeling for the disjoint union. The smallest number of isolated vertices that must be added to a graph H to achieve a sum graph is called the sum number of H; it is denoted by σ(H). A sum labeling which realizes as a sum graph is called an optimal sum labeling of H.In this paper we survey a new type of labeling based on summation, the exclusive sum, labeling. A sum labeling L is called exclusive sum labeling with respect to a subgraph H of G if L is a sum labeling of G where H contains no working vertex. The exclusive sum number ∊(H) of a graph H is the smallest number r such that there exists an exclusive sum labeling L which realizes as a sum graph. A labeling L is an optimal exclusive sum labeling of a graph H if L is a sum labeling of H ≼ K∊(H) and H contains no working vertex.

On square sum graphs

Let G = (V, E) be a (p, q)-graph and let f: V(G) → {0, 1, 2, …, p - 1} be a bijection. We define f* on E(G) by f* (uv) = [f(u)]2 + [f(v)]2. If f* is injective on E(G), then f is called a square sum labeling. If f* (E) consists of the first q consecutive integers of the form a2 + b2, a ≤ p – 1, b ≤ p – 1, a ≠ b, then f is said to be a strongly square sum labeling of G. The graph G is said to be a square sum graph (strongly square sum graph) if G admits a square sum (strongly square sum) labeling. In this paper we initiae a study of graphs which are square sum and strongly square sum.

A sum labelling for the generalised friendship graph

We provide an optimal sum labelling scheme for the generalised friendship graph, also known as the flower (a symmetric collection of cycles meeting at a common vertex) and show that its sum number is 2.

Totally magic graphs

A total labeling of a graph with v vertices and e edges is defined as a one-to-one map taking the vertices and edges onto the integers 1,2,…, v+ e. Such a labeling is vertex magic if the sum of the label on a vertex and the labels on its incident edges is a constant independent of the choice of vertex, and edge magic if the sum of an edge label and the labels of the endpoints of the edge is constant. In this paper we examine graphs possessing a labeling that is simultaneously vertex magic and edge magic. Such graphs appear to be rare.

Super-Magic Complete k-Partite Hypergraphs

We deal with complete k-partite hypergraphs and we show that for all k≥2 and n≠2,6 its hyperedges can be labeled by consecutive integers 1,2,…,nk such that the sum of labels of the hyperedges incident to (k−1) particular vertices is the same for all (k−1)-tuples of vertices from (k−1) independent sets.

Connected graphs which are not mod sum graphs

In this paper we prove that no wheel with the exception of W 4 can be a mod sum graph. We also give the unique (up to multiplication by positive integers) mod sum labelling of W 4. We also prove that the symmetric complete bipartite graph K n. n is not a mod sum graph.

Distribution of the Height of a Random Tree with Labeled Edges

A random genealogical tree of n generations of a supercritical Galton-Watson branching process with generating function $h(s), h(0) = 0, h'(1) = A > 1$, is considered; the tth level of vertices of the tree corresponds to the particles of the tth generation. Edges of the tree are labelled by independent and identically distributed random variables $\{\xi _\alpha\}$ with distribution function $G(x) = {\bf P}\{ \xi _\alpha \leqq x\} $. The weight of the path from the root to a vertex of the nth level is defined as the sum of labels $\xi _\alpha $ of all the edges of this path. The height $\eta _n $ of the tree is the maximum weight over all such paths. It is shown that the distribution function $F_n (x) = {\bf P}\{ \eta _n < x\} $ satisfies the recursion relation \[ F_{n + 1} (x ) = h\left({F_n \ast G (x )} \right),\quad n \geqq 1,\quad F_1 (x ) = h(G (x )). \] It is proved that if $G(x)$ is a bounded lattice distribution with $G(x_0 ) = 1$ and $q = G(x_0 ) - G(x_0 - 1) > 0$, $Aq > 1$ then $\lim _{n \to \infty } {\bf P} \{ nx_0 - \eta _n = kl\}$ exists for any $k = 0,1,2, \ldots $, where l is the lattice span.