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An $r$-uniform hypergraph is linear if every two edges intersect in at mostone vertex. The $r$-expansion $F^{r}$ of a graph $F$ is the $r$-uniformhypergraph obtained from $F$ by enlarging each edge of $F$ with a vertex subsetof size $r-2$ disjoint from the vertex set of $F$ such that distinct edges areenlarged by disjoint subsets. Let $ex_{r}^{lin}(n,F^{r})$ and$spex_{r}^{lin}(n,F^{r})$ be the maximum number of edges and the maximumspectral radius of all $F^{r}$-free linear $r$-uniform hypergraphs with $n$vertices, respectively. In this paper, we present the sharp (or asymptotic)bounds of $ex_{r}^{lin}( n,F^{r})$ and $spex_{r}^{lin}(n,F^{r})$ byestablishing the connection between the spectral radii of linear hypergraphsand those of their shadow graphs, where $F$ is a $(k+1)$-color critical graphor a graph with chromatic number $k$.

Linear spectral Turan problems for expansions of graphs with given chromatic number