Structures in topological recursion relations

In this paper, we study the basic structures of degree-$g$ topological recursion relations on the moduli space of curves $\overline{\mathcal{M}}_{g,n}$: (i) The coefficient of the bouquet class on $\overline{\mathcal{M}}_{g,n}$, which gives the answer to a conjecture of T. Kimura and X. Liu; (ii) Linear relations among the coefficients of certain rational tails locus of $\overline{\mathcal{M}}_{g,n}$. Three applications of topological recursion relations will be discussed: (i) Coefficients of universal equations for Gromov-Witten invariants for any smooth projective variety; (ii) The coefficient of the bouquet class in the double ramification formula of the top Hodge class $\lambda_g$; (iii) A new recursive formula for computing the intersection numbers on the moduli space of stable curves.