Let $\alpha \left(n\right)$ be the least number $k$ for which there exists a simple graph with $k$ vertices having precisely $n\ge 3$ spanning trees. Similarly, define $\beta \left(n\right)$ as the least number $k$ for which there exists a simple graph with $k$ edges having precisely $n\ge 3$ spanning trees. As an $n$-cycle has exactly $n$ spanning trees, it follows that $\alpha \left(n\right),\beta \left(n\right)\le n$. In this paper, we show that $\alpha \left(n\right)\le \frac{1}{3}(n+4)$ and $\beta \left(n\right)\le \frac{1}{3}(n+7)$ if and only if $n\notin \{3,4,5,6,7,9,10,13,18,22\}$, which is a subset of Euler’s idoneal numbers. Moreover, if $n\neg \equiv 2(mod3)$ and $n\ne 25$ we show that $\alpha \left(n\right)\le \frac{1}{4}(n+9)$ and $\beta \left(n\right)\le \frac{1}{4}(n+13).$ This improves some previously estabilished bounds.

An extended tree of a graph is a certain analogue of spanning tree. It is defined by means of vertex splitting. The properties of these trees are studied, mainly for complete graphs.

Gutman and Wagner proposed the concept of the matching energy which is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph. And they pointed out that the chemical applications of matching energy go back to the 1970s. Let T be a tree with n vertices. In this paper, we characterize the trees whose complements have the maximal, second-maximal and minimal matching energy. Furthermore, we determine the trees with edge-independence number p whose complements have...