On the Irregularity of $$\pi $$-Permutation Graphs, Fibonacci Cubes, and Trees

Abstract

The irregularity of a graph G is the sum of $$|\mathrm{deg}(u) - \mathrm{deg}(v)|$$ over all edges uv of G. In this paper, this invariant is considered on $$\pi $$ -permutation graphs, Fibonacci cubes, and trees. An upper bound on the irregularity of $$\pi $$ -permutations graphs is given, and $$\pi $$ -permutation graphs that attain the equality are characterized. The concept of the irregularity is extended to arbitrary edge subsets and applied to permutation edges of $$\pi $$ -permutation graphs. An exact formula for the irregularity of Fibonacci cubes is proved. An upper bound on the irregularity of trees in terms of the diameter is given, and trees that attain the equality are characterized.