# General questions on Cayley graphs

In Graph Theory mainly in Cayley graphs there are four general questions " according to Audery Terras" : 'Suppose A is the adjacency operator of a connected regular (undirected) graph $X$ of degree $k$ (without multiple edges). Let $spec(A)$ denote the spectrum of $A$, that is, the set of all eigenvalues of $A$. Let $d$ be the diameter of $X$ and $g$ be the girth.

Question 1. Is $X$ Ramanujan, that is, if $λ ∈spec(A)$, $|λ|≠k$ does $λ$ satisfy $|λ|≤ 2(k-1)^{1/2}$?

Question 2. Is $0∈ spec(A)$ or, equivalently, is $A$ invertible?

Question 3. Can we bound the diameter $d$?

Question 4. Can we bound the girth $g$?

As I am new in this field of mathematics, my question is: are there more important or maybe new questions that researcher can ask or for example analogue some questions?

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