On the minimum reformulated Albertson Index of fixed-order trees and unicyclic graphs with a given maximum degree

The Albertson index (Formula presented.), traditionally based on vertex degrees, is defined as the sum of the absolute value of the differences in degrees between adjacent vertices. In this study, we introduce an edge version of this variant termed the reformulated Albertson index (Formula presented.), where the sum is taken over the absolute value of the differences in degrees between adjacent edges. We explore this index and establish a sharp lower bound for trees and unicyclic graphs, expressed in terms of the maximum degree and the number of pendant vertices attached to the vertex of the maximum degree. Additionally, we derive some upper bounds for the (Formula presented.) in terms of order, size, minimum, and maximum degree.