The Square Graph of a Cycle Graph - Saha, Nabajit; Das, Manmohan (Communications in Mathematics and Applications, 2025)
The square graph of a cycle graph \(C_{n}\) is a graph consisting of the same set of vertex as in \(C_{n}\) and two different vertices are considered adjacent if their distance in \(C_{n}\) is at most two. We denote the square graph of a cycle graph \(C_{n}\) by \(C^2_{n}\). In this article, we first show that the square graph \(C^2_{n}\) is a 4-regular graph for \(n≥5\) and then discuss various graphical properties such as the spectrum, Laplacian spectrum of \(C^2_{n}\), tree number, energy and, finally, the vertex coloring of \(C^2_{n}\).
Keywords: Square graph; Spectrum; Tree number; Energy of a graph; Vertex coloring