Semigroups of transformations and endomorphisms have emerged as powerful algebraic frameworks to elucidate the underlying structures of graphs. By harnessing the principles of semigroup theory, ...

Semigroups of transformations and endomorphisms have emerged as powerful algebraic frameworks to elucidate the underlying structures of graphs. By harnessing the principles of semigroup theory ...

\(y = x^2 + a\) represents a translation parallel to the \(y\)-axis of the graph of \(y = x^2\). If \(a\) is positive, the graph translates upwards. If \(a\) is ...

A translation is a movement of the graph either horizontally parallel to the \(x\)-axis or vertically parallel to the \(y\)-axis. The graph of \(f(x) = x^2\) is the same as the graph of \(y = x^2\).

Irl C. Bivens, The Linear Transformation Associated with a Graph, The College Mathematics Journal, Vol. 24, No. 1 (Jan., 1993), pp. 76-78 ...