Extremal t-apex trees with respect to matching energy

The matching energy of a graph is defined as the sum of the absolute values of the zeros of its matching polynomial. For any integer t≥1, a graph G is called t-apex tree if there exists a t-set X⊆V(G) such that G - X is a tree, while for any Y⊆V(G) with |Y|<t, G - Y is not a tree. Let Tt(n) be the set of t-apex trees of order n. In this article, we determine the extremal graphs from Tt(n) with minimal and maximal matching energies, respectively. Moreover, as an application, the extremal cacti of order n and with s cycles have been completely characterized at which the minimal matching energy are attained.