Some properties of stepwise irregular graphs

Bera, Somnath (Huazhong University of Science and Technology)
Paul, Prithwineel (Southwest Jiaotong University)
Subramanian, K.G. (Visiting Professor (Honorary) Liverpool Hope University)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 14, No 1 (2026): Electronic Journal of Graph Theory and Applications

Graphs in which the absolute difference between the degrees of any two adjacent vertices is exactly one, are called stepwise irregular (SI) graphs. We establish several properties of SI graphs. In particular, we show that SI graphs of different order and cyclomatic numbers can be constructed from an SI graph with a vertex of degree 1 or 2. Necessary conditions and sufficient conditions for a degree sequence to be SI graphic are obtained. Moreover, a necessary condition comparing the sum of the terms of a partition of SI graph and its conjugate partition is obtained. Properties of SI graphs under certain elementary graph operations are also investigated.