TOPICS

# Irregular Graph

The term "irregular graph" may be used to refer to a graph that is not regular.

However, a different definition considers an edge-weighted graph in which each edge is assigned a positive integer weight and the degree of a vertex is defined as the sum of the weights of its incident edges. Such a graph may be termed irregular if all the vertices have distinct degrees (Faudree et al. 1989, Przybylo 2024).

Highly Irregular Graph, Irregularity Strength, Regular Graph, Strongly Regular Graph, Weakly Regular Graph

## Explore with Wolfram|Alpha

More things to try:

## References

Chartrand, G.; Erdős, P.; and Oellermann, O. R. "How to Define an Irregular Graph." College Math. J. 19, 36-42, 1988.Faudree, R.; Schelp, R.; Jacobson, M.; and Lehel, J. "Irregular Networks, Regular Graphs and Integer Matrices With Distinct Row and Column Sums." Disc. Math. 76,) 223-240, 1989.Przybylo, J. "The Irregularity Strength of Dense Graphs--On Asymptotically Optimal Solutions of Problems of Faudree, Jacobson, Kinch and Lehel." 13 Jun 2024. https://arxiv.org/abs/2406.09584.

## Cite this as:

Weisstein, Eric W. "Irregular Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IrregularGraph.html