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, ...

A translation is a shift of the graph either horizontally parallel to the \(x\)-axis or vertically parallel to the \(y\)-axis. If \(f(x) = x^2\), then \(f(x) + a = x^2 + a\). The value of \(a\) ...

If \(f(x) = x^2\), then \(af(x) = a(x^2)\). This tells us that we need to multiply each of the \(y\) coordinates on the graph by \(a\) in order to stretch the original graph. Looking at some ...

The field of model checking has evolved into a fundamental technique for verifying the correctness of software and hardware systems. At its core, model checking systematically explores the state space ...