A d-handicap distance antimagic labeling of a graph G = (V, E) with n vertices is a bijection f : V → {1, 2, . . . , n} with the property that f(xi) = i and the sequence of weights w(x1), w(x2), . . . , w(xn) (where w(xi) = ∑ xixj ∈E f(xj)) forms an increasing arithmetic progression with common difference d. A graph G is a d-handicap distance antimagic graph if it allows a d-handicap distance a...

A distance magic labeling of a graph G = (V, E) of order n is a bijection l : V → {1, 2, . . . , n} with the property that there is a positive integer k (called magic constant) such that w(x) = k for every x ∈ V . If a graph G admits a distance magic labeling, then we say that G is a distance magic graph. In the case of non-regular graph G, the problem of determining whether there is a distance...

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For a graph property P, the edit distance of a graph G from P, denoted EP(G), is the minimum number of edge modifications (additions or deletions) one needs to apply to G in order to turn it into a graph satisfying P. What is the furthest graph on n vertices from P and what is the largest possible edit distance from P? Denote this maximal distance by ed(n,P). This question is motivated by algor...

An (r, n)-split coloring of a complete graph is an edge coloring with r colors under which the vertex set is partitionable into r parts so that for each i, part i does not contain K, in color i. This generalizes the notion of split graphs which correspond to (2, 2)-split colorings. The smallest N for which the complete graph Ks has a coloring which is not (r,n)-split is denoted by f,.(n). Balan...

Let G = (V,E) be a graph of order n. A bijection f : V → {1, 2, . . . , n} is called a distance magic labeling of G if there exists a positive integer k such that ∑ u∈N(v) f(u) = k for all v ∈ V, where N(v) is the open neighborhood of v. The constant k is called the magic constant of the labeling f . Any graph which admits a distance magic labeling is called a distance magic graph. In this pape...

We generalize the concept of a 2-coloring of a graph to what we call a semibalanced coloring by relaxing a certain discrepancy condition on the shortest-paths hypergraph of the graph. Let G be an undirected, unweighted, connected graph with n vertices and m edges. We prove that the number of different semi-balanced colorings of G is: (1) at most nþ 1 if G is bipartite; (2) at most m if G is non...

A signed graph has edge weights drawn from the set $\{+1,-1\}$, and is termed sign-balanced if it equivalent to an unsigned under operation of sign switching; otherwise called sign-unbalanced. nut a one dimensional kernel with corresponding eigenvector that full. In this paper we generalise notion graphs graphs. Orders for which regular exist were determined recently degrees up $11$. By extendi...