DIAMETER OF COMMUTING GRAPHS OF SYMPLECTIC ALGEBRAS

Let $F$ be an algebraically closed field of characteristic $0$ and let $\operatorname{sp}(2l,F)$ be the rank $l$ symplectic algebra of all $2l\times 2l$ matrices $x=\big(\!\begin{smallmatrix}A & B\\ C & -A^{t}\end{smallmatrix}\!\big)$ over $F$ , where $A^{t}$ is the transpose of $A$ and $B,C$ are symmetric matrices of order $l$ . The commuting graph $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ of $\operatorname{sp}(2l,F)$ is a graph whose vertex set consists of all nonzero elements in $\operatorname{sp}(2l,F)$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=yx$ . We prove that the diameter of $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ is $4$ when $l>2$ .