Super edge-magic total strength of some unicyclic graphs

TL;DR

This paper investigates the super edge-magic total strength of certain unicyclic graphs, proposing a conjecture that relates this strength to the graph's parameters, with evidence supporting the proposed formula.

Contribution

It introduces a new conjecture linking super edge-magic total strength to unicyclic graph parameters and provides partial evidence for this relationship.

Findings

Proposes a conjecture for super edge-magic total strength of unicyclic graphs.

Provides evidence supporting the conjecture for specific graph classes.

Suggests a formula: strength equals 2q + (n+3)/2 for certain unicyclic graphs.

Abstract

Let $G$ be a finite simple undirected $(p,q)$-graph, with vertex set $V(G)$ and edge set $E(G)$ such that $p=|V(G)|$ and $q=|E(G)|$. A super edge-magic total labeling $f$ of $G$ is a bijection $f\colon V(G)\cup E(G)\longrightarrow \{1,2,\dots , p+q\}$ such that for all edges $u v\in E(G)$, $f(u)+f(v)+f(u v)=c(f)$, where $c(f)$ is called a magic constant, and $f(V(G))=\{1,\dots , p\}$. The minimum of all $c(f)$, where the minimum is taken over all the super edge-magic total labelings $f$ of $G$, is defined to be the super edge-magic total strength of the graph $G$. In this article, we work on certain classes of unicyclic graphs and provide shreds of evidence to conjecture that the super edge-magic total strength of a certain family of unicyclic $(p,q)$-graphs is equal to $2q+\frac{n+3}{2}$.