A Study on Set-Valuations of Signed Graphs

TL;DR

This paper explores set-valuations of signed graphs, defining new labeling concepts based on set theory, and investigates properties of signed graphs that admit these set-valuations.

Contribution

It introduces the concept of set-valuations for signed graphs and analyzes their properties, extending existing graph labeling theories.

Findings

Characterization of signed graphs with set-valuations

Conditions for the existence of set-indexers in signed graphs

Properties of signed graphs admitting specific set-valuations

Abstract

Let $X$ be a non-empty ground set and $\mathcal{P}(X)$ be its power set. A set-labeling (or a set-valuation) of a graph $G$ is an injective set-valued function $f:V(G)\to \mathcal{P}(X)$ such that the induced function $f^\oplus:E(G) \to \mathcal{P}(X)$ is defined by $f^\oplus(uv) = f(u)\oplus f(v)$, where $f(u)\oplus f(v)$ is the symmetric difference of the sets $f(u)$ and $f(v)$. A graph which admits a set-labeling is known to be a set-labeled graph. A set-labeling $f$ of a graph $G$ is said to be a set-indexer of $G$ if the associated function $f^\oplus$ is also injective. In this paper, we define the notion of set-valuations of signed graphs and discuss certain properties of signed graphs which admits certain types of set-valuations.