Tricyclic graphs with the second largest distance eigenvalue less than −12

Let G be a simple connected graph with vertex set V(G)={v₁,v₂,…,v_n}. The distance d_G(v_i,v_j) between two vertices v_i and v_j of G is the length of a shortest path between v_i and v_j. The distance matrix of G is the matrix D(G)=(dG(vi,vj))_n×n. The second largest distance eigenvalue of G is the second largest eigenvalue of D(G). Guo and Zhou (2024) proved that any connected graph with the second largest distance eigenvalue less than −12 is chordal, and characterized all bicyclic graphs and split graphs with the second largest distance eigenvalue less than −12. In this paper, we characterize all tricyclic graphs with the second largest distance eigenvalue less than −12.