فهرست نشریات علمی دسترسی آزاد ایران

‎The Gutman index is a degree-distance-based topological descriptor of connected graphs‎. ‎In this paper‎, ‎we derive explicit analytic expressions for its expected value in polyomino chains built by sequentially attaching square tiles via one of two fixed local connection modes‎. ‎This expectation is expressed as a cubic polynomial in the number of tiles $n$‎. ‎We then identify which attachment patterns yield the extremal (maximum and minimum) values and compute the overall average of the Gutman index across all polyomino chains of length $n$‎. ‎These results enhance the topological analysis of square-tiled networks with applications in chemical graph theory‎, ‎polymer science‎, ‎and materials design‎.