Let $c_i$ be the number of complete subgraphs of size $i$ in a graph $G$. I learned the following result:

If $G$ has $n$ vertices and is connected and chordal, that is, $G$ does not have induced cycles of length at least 4, then 
$$
\sum_{i=1}^n (-1)^{i+1} c_i = 1
$$

See for example Thm 4.1 (together with the fact that $\lambda=1$ is a root after Theorem 3.1) in this [paper][1], or Prop 2.3 of [this paper][2].

My question is the following:

1. Is there an elementary proof of this fact without resorting to topological arguments?

2. Is there a generalization of this to k-chordal graphs?


  [1]: http://www.sciencedirect.com/science/article/pii/S0012365X02005009
  [2]: https://arxiv.org/pdf/1004.3416.pdf