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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 andjoining 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 notcontaining $\widehat P_4$ is $\left\lfloor n^2/8\right\rfloor$. Our method usessimple induction along with computer programming to prove a base case of theinduction hypothesis.

Exact generalized Tur\'an number for $K_3$ versus suspension of $P_4$