Some families of digraphs determined by the complementarity spectrum.

In this paper we study seven families of digraphs, and we determine whether digraphs in these families can be determined by their spectral radius. These seven families have been characterized as the only families of digraphs with exactly three complementarity eigenvalues [D. Bravo et al., J. Algebr. Comb. 57, No. 4, 1173–1193 (2023; Zbl 1516.05076)], and therefore our results have consequences in this context, showing which families can be determined by the complementarity spectrum. As a particular case, we prove that the θ-digraphs can be characterized by the spectral radius, extending some recent results on this family [H. Lin and J. Shu, Linear Algebra Appl. 436, No. 7, 2524–2530 (2012; Zbl 1238.05163)]