Abstract
For a group $G$ with $G$-conjugacy class of involutions $X$, the local fusion graph $\mathcal{F}(G,X)$ has $X$ as its vertex set, with distinct vertices $x$ and $y$ joined by an edge if, and only if, the product $xy$ has odd order. Here we show that, with only three possible exceptions, for all pairs $(G,X)$ with $G$ a sporadic simple group or the automorphism group of a sporadic simple group, $\mathcal{F}(G,X)$ has diameter $2$.
Highlights
Suppose that G is a finite group with X a G-conjugacy class of involutions
The local fusion graph, F(G, X), is the graph whose vertex set is X with distinct vertices x and y joined by an edge whenever xy has odd order
There, examples are given of groups which have local fusion graphs whose diameter can be arbitrarily large
Summary
Suppose that G is a finite group with X a G-conjugacy class of involutions (that is, a G-conjugacy class of elements of order 2). There, examples are given of groups which have local fusion graphs whose diameter can be arbitrarily large. For (G, X) = (M, 2B) it can be shown that the diameter of the local fusion graph is at most 6 This follows from [13], where it is shown that the commuting involution graph of M on the 2B conjugacy class has diameter 3, when combined with the observation that two commuting 2B involutions are distance 2 apart in F(M, 2B). This bound is almost certainly not the best possible. Note that Lemma 3 is best possible, as the example of dumbbell graphs attest
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