  Let $P_4$ denote the path graph on $4$ vertices. The suspension of $P_4$,
denoted by $\widehat P_4$, is the graph obtained via adding an extra vertex and
joining it to all four vertices of $P_4$. In this note, we demonstrate that for
$n\ge 8$, the maximum number of triangles in any $n$-vertex graph not
containing $\widehat P_4$ is $\left\lfloor n^2/8\right\rfloor$. Our method uses
simple induction along with computer programming to prove a base case of the
induction hypothesis.

Combinatorics