PBIB Designs and Association Scheme Arising from Minimum Total Dominating Sets of Non Square Lattice Graph

Square lattice graphs L_2 (n) with the parameters (n^2, 2(n-1), n-2, 2) are strongly regular and are unique for all n except n=4. however for n=4, we have two non-isomorphic strongly regular graphs. The non-lattice graph with parameters (16,6,2,2 ) is known as Shrikhande graph. In this paper we show that every minimum total dominating set in Shrikhande graph induces two K_2s. Further we establish that these classes of minimum total dominating sets of Shrikhade graph form Partially Balanced Incomplete Block Designs with the parameters (16, 44, 11, 4, λi, i=1or2, λij=3or4).