Minimum Domination Reciprocal Mean Square Index for Fixed-Order Graphs

Determine the minimum of the Domination Reciprocal Mean Square index among connected graphs, or among trees, of a given order.

Background

The paper introduces the Domination Reciprocal Mean Square index (DRMS), defined by summing the reciprocal of the squared domination degrees at the endpoints of each edge. It establishes upper and lower bounds, equality characterizations, and closed-form evaluations for several graph families, but does not determine the minimum possible value for a fixed graph order.

The authors explicitly identify the minimization problem over connected graphs, with the corresponding restricted problem over trees, as unresolved and leave it for future work. The problem concerns extremal graph theory for the newly introduced domination-degree-based index.