Eigenvalue bounds and girths of graphs of finite, upper half-planes (Q1891175)

For each odd prime power \(q= p^ r\) we will investigate \(q- 2\) different Cayley graphs called finite, upper half-planes over \(F_ q\). We define a finite, upper half-plane by \[ H_ q= \{x+ y\sqrt d\mid x, y\in F_ q, y\neq 0\} \] where \(F_ q\) is the finite field with \(q\) elements and \(d\) is not a square in \(F_ q\). We define a distance \(k\) between points \(z\) and \(w\in H_ q\) by \[ k(z, w)= {N(z- w)\over (\text{Im } z)(\text{Im } w)} \] where \(Nz= z\overline z\) and \(\overline z= x- y\sqrt d\) and \(\text{Re }z= x\) and \(\text{Im }z= y\). We define a graph, \(X_ q(d,a)\), by letting the elements of \(H_ q\) be the vertices of the graph and defining an edge between \(z\) and \(w\) when \(k(z, w)= a\) for a fixed \(a\in F_ q- \{0\}\). We consider the origin to be the point \(\sqrt d\). We call \(H_ q(d, a)\), the finite upper half- plane depending on a fixed \(a\) and \(d\). We first concern ourselves with whether the eigenvalues, \(\lambda\), of the adjacency matrices of the graphs satisfy the Ramanujan bound \(|\lambda|\leq \sqrt q\). Since the graphs are of degree \(q+1\), the paper shows a method to use the representations for the additive and multiplicative groups of each \(F_ q\) to find the smaller associated isospectral matrices. We then find the eigenvalues of the isospectral matrices. A computer program has verified the Ramanujan bound for most of the graphs up to the prime power \(3^ 5\). We next concern ourselves with the girth of the graphs. This paper shows that the girths are either 3 or 4 and shows that the girth is 3 if \(a= 2d\) and \(q\equiv 3\pmod 4\) or if \(a\) and \(a- 3d\) are squares in \(F_ q\). The girth is 4 if \(a= 2d\) and \(q\equiv 1\pmod 4\).