Is the following bound on the largest eigenvalue of a partitioned graph known? It seems like it should be but I am not able to find a reference to it.
Given a graph $G=(V,E)$ whose vertices are partitioned into $V=V_1 \cup V_2$ let $d_i(u)$ be the internal degree of $u$, the number of neighbors of $u$ that are in the same partition and let $d_o(u)$ be the outer degree of $u$, the number of neighbors of $u$ not in the same partition. So $d_i(u)+d_o(u)=d(u)$ for any vertex $u$. Now for either partition $S$ call $d_i(S)$, the internal degree of $S$, the largest internal degree amongst the vertices of $S$. That is, $$d_i(S)= max_{u \in S} d_i(u).$$ And likewise for the outer degree of a partition: $$d_o(S)= max_{u \in S} d_o(u).$$
If my sums are correct I bound the spectral radius of $A(G)$, the adjacency matrix of $G$ by $$\rho(G) \leq \max\{d_i(V_1),d_i(V_2)\} + \sqrt{d_o(V_1) d_o(V_2)}.$$
My question is whether, or not, this bound is already known -- assuming I haven't messed up :-). I cannot find any mention of it in the obvious places (Godsil/Royle or Brouwer/Haemers).
Some comments. The bound generalizes two common special cases: 1) if $V_1, V_2$ is a bipartition then $d_i(V_1)=d_i(V_2)=0$ and we get a version of the familiar bipartite graph bound; 2) if $V_2=\emptyset$ then we get the familiar maximum degree bound; Brouwer/Haemers discuss something similar (using interlacing and almost equitable partitions) but the emphasis there is on average degree, not maximum.
Many thanks.