What is the meaning of this curve in the graph of equation x^x=y^y?

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Using Desmos I plotted the graph of equation $x^x = y^y$ which looks like this
Graph of x^x = y^y
According to my understanding the solution to this Equation should have been only $y=x$ but the curve seems to prove me wrong.
So, what is the reason of that curve from $(0,1)$ to $(1,0)$ and what is the value of $x$ where the Curve intersects with line (I mean how can we calculate its value algebraically)?