Let $G$ be a graph with adjacency matrix $A_G$ and $H$ be an induced subgraph of $G$. Let $H^*$ be a graph with same number of vertices as $H$ such that $\rho(A_H) \leq \rho(A_{H^*})$, where $\rho(A)$ denotes the spectral radius of $A$. Consider the graph $G^*$ by deleting the induced subgraph $H$ and replacing it by $H^*$.
Can we conclude that $\rho(G) \leq \rho(G^*)?$
If it is an existing result/theorem can you provide some refernce?
This statement does not hold. Here is an example built by taking $H,H^*$ to be isospectral graphs. Let $x = (1,1,1,1,1)$. Take $$ A_{H} = \pmatrix{ 0 &1 &0 &1 &0\\ 1 &0 &1 &0 &0\\ 0 &1 &0 &1 &0\\ 1 &0 &1 &0 &0\\ 0 &0 &0 &0 &0 }, \qquad A_{H^*} = \pmatrix{ 0 & 1 & 1 & 1 & 1\\ 1 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0\\ },\\ A_G = \pmatrix{A_H & x\\x^T & 0}, \quad A_{G^*} = \pmatrix{A_{H^*} & x\\ x^T & 0}. $$ Then we have $2 = \rho(H) \leq \rho(H^*) = 2,$ but $\rho(G) \approx 3.67 > \rho(G^*) \approx 3.37$.