Reciprocal Irregularity Under Partial Edge Subdivision

Edge subdivision is a basic graph transformation, but different degree-irregularity measures need not react to it in the same way. This paper determines the exact response of reciprocal irregularity when an arbitrary set of edges is subdivided, with each selected edge allowed to receive any positive number of new degree-two vertices. A local formula is obtained from the endpoint degrees, yielding an additive law for partial subdivision, a complete equality characterization, invariance with respect to repeated subdivision along the same original edge, and an immediate fixed-budget edge-selection principle. Reciprocal irregularity is shown never to decrease; outside the exceptional graph K2, a selected set produces a strict increase precisely when it contains an edge whose two original endpoints both have degree at least three. A comparative theorem proves that the Albertson-irregularity change equals 2 min{d(u), d(v)} times the reciprocal change, so the two measures have identical local sign behavior. In contrast, the sigma irregularity changes by 2(d(u)-2)(d(v)-2) and therefore decreases on pendant edges incident with a vertex of degree at least three. Extensive exact-arithmetic Graph Atlas verification found no counterexample to the stated identities and classifications.