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

If $G$ is 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, or Prop 2.3 of this paper.

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?

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, or Prop 2.3 of this paper.

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?