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?
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 $d$.