"The" Y-graph is another term used to refer to a claw
graph.

The term "Y-graph" is also used to refer to a graph expansion with the Y graph as its base (e.g., Horton and Bouwer 1991). There
are exactly four graph expansions with Y-graph base that are symmetric
(Biggs 1993, p. 147).

Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, p. 147,
1993.Bouwer, I. Z.; Chernoff, W. W.; Monson, B.; and Star,
Z. The Foster Census. Charles Babbage Research Centre, 1988.Horton,
J. D. and Bouwer, I. Z. "Symmetric Y-Graphs and H-Graphs." J.
Combin. Th. Ser. B53, 114-129, 1991.