# What does edge-transitive imply for the adjacency matrix of graph?

I'm trying to prove that if a matrix $$G$$ is edge transitive, we can say that

$$\vartheta(G) = \frac{n\lambda_{min}(A_G)}{\lambda_{max}(A_G) -\lambda_{min}(A_G)}$$

Where $$\vartheta(G)$$ is defined in multiple ways:

• $$\min\{ t \mid C-J\succeq 0$$ , $$C_{ii} = t$$ $$\forall i\in V$$, $$C_{ij}=0$$ for $$\{i,j\}\in \bar{E}\}$$ (With $$J$$ matrix of all ones)
• $$\max\{ \Sigma_{i\in V} Y_{ii} \mid Y\succeq 0$$, $$Y_{00} =1, Y_{ij}=0$$ for $$\{i,j\}\in E$$, $$Y_{0i}=Y_{ii}\}$$

I have already shown for $$k$$-regular graphs $$G$$ that $$\vartheta(G) \leq min_{x\in\mathbb{R}} \lambda_{max}(J-xA_G)$$

So, i'm thinking the answer should be somewhere in the adjacency matrix.

Any wisdom that can help me get this show on a road?