The reflexive edge strength on some almost regular graphs

Area of Science:

• Graph Theory
• Combinatorics
• Discrete Mathematics

Background:

• Introduces total k-labeling, assigning numbers to graph edges and vertices.
• Defines edge irregular reflexive k-labeling, where adjacent edges and incident edges/vertices have distinct sums.
• Establishes reflexive edge strength (χ'(G)) as the minimum k for such a labeling.

Purpose of the Study:

• Determine the reflexive edge strength for specific graph classes.
• Investigate graphs exhibiting near-regularity in vertex degrees.

Main Methods:

• Utilized graph labeling techniques.
• Applied combinatorial methods to analyze graph properties.
• Focused on graphs with vertex degrees showing almost regularity.

Main Results:

• Derived the reflexive edge strength for graphs with vertex degrees exhibiting almost regularity.
• Provided specific values or bounds for χ'(G) in these graph classes.

Conclusions:

• The study contributes to understanding graph labeling invariants.
• Results offer insights into the structure of graphs with near-regular vertex degrees.