Is every symmetric graph a strongly regular graph?

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I have a bit of a problem to develop an intuition about symmetric graphs (I only consider in the following connected simple graphs)... For example cycles (but only up to 5 vertices) and complete graphs are symmetric and strongly regular. But is every symmetric graph also strongly regular?

Counter-examples or some intuitive explanation is welcome. Complicated proofs are maybe beyond what I can understand at the moment. Unfortunarely, also I am not very strong in group theory.

I know that every symmetric graph is regular.

Greets, Lando

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1 Answer

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Wolfram’s article on symmetric graphs shows $4$ connected, symmetric, simple graphs on $6$ vertices; Wolfram’s article on strongly regular graphs shows only $3$ connected, strongly regular, simple graphs on $6$ vertices. Thus, there must be a symmetric graph that isn’t strongly regular. From the illustrations it appears that $C_6$ and $K_{3,3}$ are symmetric but not strongly regular, and the graph obtained from $C_6$ by adding the $3$ edges between diametrically opposite vertices is strongly regular but not symmetric.

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