Suppose $G$ is a 2-connected graph of order $n$ with diameter $d\geq 2$. We prove that \[ |E(G)|\geq\frac{dn-2d-1}{d-1}. \] We also characterize the extremal graphs for $d\geq 5$.

Let $F=\langle a,b\rangle$ be the free group generated by $a,b$. Let $\phi\in\mathrm{Hom}(F,SL(2,\mathbf{C}))$ be a homomorphism from $F$ to $SL(2,\mathbf{C})$. Define $T(\phi)=(\mathrm{tr}\,\phi(a),\mathrm{tr}\,\phi(b),\mathrm{tr}\,\phi(ab))$, where $\mathrm{tr}\,A$ stands for the trace of the matrix $A$. Let $\sigma\in\mathrm{Aut}F$. Then from [2, 12, 4], there exists a unique polynomial map $\Phi_{\sigma}\in(\mathbf{Z}[x,y,x])^3$, such that \[ \mathrm{tr}\,\phi(\sigma(a)),\mathrm{tr}\,\phi(\sigma(b)),\mathrm{tr}\,\phi(\sigma(ab)))=\Phi_{\sigma}(\mathrm{tr}\,\phi(a),\mathrm{tr}\,\phi(b),\mathrm{tr}\,\phi(ab)) \] with $x=\mathrm{tr}\,\phi(a),y=\mathrm{tr}\,\phi(b),z=\mathrm{tr}\,\phi(ab)$, and there exists a unique polynomial $Q_{\sigma}$, such that $\lambda\circ\Phi_{\sigma}=\lambda\cdot Q_{\sigma}$, where $\lambda(x,y,z)=x^2+y^2+z^2-xyz-4$. In this paper, we will show that $\sigma\in\mathrm{Aut}F$ if and only if $Q_{\sigma}(2,2,z)\equiv 1$, and that this result cannot be improved.

In this paper we establish strong ergodic theorems for non-Lipschitzian mappings of asymptotically nonexpansive type in uniformly convex Banach spaces.

We show that if a differentiable map of a smooth manifold has a non-atomic ergodic hyperbolic measure then the topological entropy is positive and the space contains a hyperbolic horseshoe. Moreover we give some relations between hyperbolic measures and periodic points for differentiable maps. These are generalized contents of the results obtained by Katok for diffeomorphisms.