Spectral Turán problems for suspensions of balanced trees

A central problem in spectral Tur\'an theory is to understand the relationship between the spectral extremal family ${\rm SPEX}(n,F)$ and the ordinary extremal family ${\rm EX}(n,F)$. For many forbidden graphs $F$, it is known that ${\rm SPEX}(n,F)\subseteq{\rm EX}(n,F)$ holds for infinitely many $n$, while only a few examples have been identified where the two families are disjoint. In this paper, we study this problem for suspensions of balanced trees. A tree is balanced if its two bipartition classes differ in size by at most one. Let $T$ be a balanced tree on $2k$ or $2k+1$ vertices and $\widehat T$ be its suspension which is obtained from $T$ by adding one new vertex adjacent to every vertex of $T$. Our first main result establishes a tight upper bound for the spectral Tur\'an number of $\widehat T$ for sufficiently large $n$ provided that $T$ satisfies some mild assumptions. Our second result determines for which integers $k$ and which non-path balanced trees $T$ on $2k$ or $2k+1$ vertices there are infinitely many integers $n$ such that ${\rm EX}(n,\widehat{T})\cap {\rm SPEX}(n,\widehat{T})=\emptyset$.