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A $(p,q)$-sigraph $S$ is an ordered pair $(G,s)$ where $G=(V,E)$ is a $(p,q)$-graph and $s$ is a function which assigns to each edge of $G$ a positive or a negative sign. Let the sets $E^{+}$ and $E^{-}$ consist of $m$ positive and $n$ negative edges of $G$, respectively, where $m+n=q$. Given positive integers $k$ and $d$, $S$ is said to be $(k,d)$-graceful if the vertices of $G$ can be labeled with distinct integers from the set $\{0,1,\cdots ,k+(q-1\left)d\right\}$ such that when each edge $uv$ of $G$ is assigned the product of its sign and the absolute difference of the integers assigned to $u$ and $v$ the...

In our earlier paper [9], generalizing the well known notion of graceful graphs, a $(p,m,n)$-signed graph $S$ of order $p$, with $m$ positive edges and $n$ negative edges, is called graceful if there exists an injective function $f$ that assigns to its $p$ vertices integers $0,1,\cdots ,q=m+n$ such that when to each edge $uv$ of $S$ one assigns the absolute difference $\left|f\right(u)-f(v\left)\right|$ the set of integers received by the positive edges of $S$ is $\{1,2,\cdots ,m\}$ and the set of integers received by the negative edges of $S$ is $\{1,2,\cdots ,n\}$. Considering the conjecture therein that all...

Let G be a graph of order n and let S be a set of positive integers with |S| = n. Then G is said to be S-magic if there exists a bijection ϕ : V (G) → S satisfying ∑x∈N(u) ϕ(x) = k (a constant) for every u ∈ V (G). Let α(S) = max{s : s ∈ S}. Let i(G) = min α(S), where the minimum is taken over all sets S for which the graph G admits an S-magic labeling. Then i(G) − n is called the distance magic index of the graph G. In this paper we determine the distance magic index of trees and complete bipartite...