On the parameters of symmetric difference bipartite graphs

The minimum degree Wiener index of a graph \( G \) is the sum of distances between all unordered pairs of vertices with minimum degree in \( G \). In this paper, we computed minimum degree Wiener index of wheel graphs, molecular graphs of graphene, oxide network etc. A class of bipartite irregular graphs named symmetric difference bipartite graph, \( SDB(n,k,i) \), is defined with two disjoint vertex partitions \( V_1 \) containing \( k \)-element subsets and \( V_2 \) containing \( (k-i) \)-element subsets and \( (k+i) \)-element subsets of \( \{1,2,\ldots,n\} \), for \( 1\leq k\leq n-1, 1\leq i \leq \min\{n-k,k-1\} \). An edge between a vertex \( U\in V_1 \) and a vertex \( W\in V_2 \) exists if their symmetric difference has exactly \( i \) elements. We determined various graph theoretic properties of symmetric difference bipartite graphs and compared its topological indices with other families of graphs.