Let \({\mathbb {K}}\) denote a field and d be a positive integer. Let V be a vector space of dimension \(d+1\) over \({\mathbb {K}}\) . A Leonard pair on V is an ordered pair \((A, A^*)\) of diagonalizable linear maps on V, with the property that each acts on an eigenbasis for the other one in an irreducible tridiagonal fashion. In this paper, we introduce the family of doubly almost-bipartite (DAB) Leonard pairs, defined as follows. Let \((A,A^*)\) denote a Leonard pair on V. Let \(\{v^*_i\}_{i=0}^d\) , denote an ordered eigenbasis for \(A^*\) on which A acts in an irreducible tridiagonal fashion. For \(0 \le i \le d\) , let \(E^*_i : V \rightarrow V\) be the \({\mathbb {K}}\) -linear map such that \(E^*_i v^*_i= v^*_i\) and \(E^*_i v^*_j = 0\) when \(j \not =i\) \((0 \le j \le d)\) . We say \((A,A^*)\) is doubly almost-bipartite (DAB) whenever A satisfies \(E^*_i A E^*_i=0 \quad \text{ if } \text{ and } \text{ only } \text{ if } \quad 1 \le i \le d-1.\) In particular, both \(E^*_0 A E^*_0\) and \(E^*_d A E^*_d\) are assumed to be non-zero. Our main result is the classification (up to isomorphism) of the DAB Leonard pairs with \(d \ge 4.\) Our work is situated within the framework of Terwilliger’s celebrated classification of Leonard pairs, and establishes that DAB Leonard pairs are exactly of the q-Racah, q-Hahn, q-Krawtchouk, or Bannai/Ito type.
