Graph theoretic interpretation of a Geometric problem

Question

I am currently reading Godsil's Algebraic graph theory for my thesis . On page 250, he says:

"The geometric problem of finding least integer d such that there are n equiangular lines in $\Bbb{R}^d$ is equivalent to the graph-theoretic problem of finding graphs X on n vertices such that the multiplicity of the least eigenvalue of $S(X)$(the corresponding Seidel matrix)is as large as possible."

I cannot see the relation between "least integer d" and "the largest multiplicity of the smallest eigenvalue". The book says that we start working with $I+\frac{1}{\alpha}S$ where $\alpha$ is the least eigenvalue of S(X) and we are assuming that the rank of the above matrix is d. How does minimizing d, maximize the multiplicity of the least eigenvalue?

Answer 1

The multiplicity $m$ of $-\alpha$ as an eigenvector of $S$ corresponds to the dimension of the nullspace of $(S+\alpha I)$, which is a scalar multiple of $(I + \frac{1}{\alpha}S)$. This means that we have $m = n-d$, where $d = \mathrm{rank}(I+\frac{1}{\alpha} S)$. Therefore having $m$ be as large as possible relies on finding the smallest possible $d$.