Kavli Affiliate: Jing Wang
| First 5 Authors: Jing Wang, Liying Kang, Yusai Xue, ,
| Summary:
For a simple graph $F$, let $mathrm{Ex}(n, F)$ and $mathrm{Ex_{sp}}(n,F)$
denote the set of graphs with the maximum number of edges and the set of graphs
with the maximum spectral radius in an $n$-vertex graph without any copy of the
graph $F$, respectively. The Tur’an graph $T_{n,r}$ is the complete
$r$-partite graph on $n$ vertices where its part sizes are as equal as
possible. Cioabu{a}, Desai and Tait [The spectral radius of graphs with no odd
wheels, European J. Combin., 99 (2022) 103420] posed the following conjecture:
Let $F$ be any graph such that the graphs in $mathrm{Ex}(n,F)$ are Tur'{a}n
graphs plus $O(1)$ edges. Then $mathrm{Ex_{sp}}(n,F)subset mathrm{Ex}(n,F)$
for sufficiently large $n$. In this paper we consider the graph $F$ such that
the graphs in $mathrm{Ex}(n, F)$ are obtained from $T_{n,r}$ by adding $O(1)$
edges, and prove that if $G$ has the maximum spectral radius among all
$n$-vertex graphs not containing $F$, then $G$ is a member of $mathrm{Ex}(n,
F)$ for $n$ large enough. Then Cioabu{a}, Desai and Tait’s conjecture is
completely solved.
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