Spectrum and Ricci Curvature on the Weighted Strong Product Graphs

Abstract

The strong product on graphs is also called the normal product or the AND product. It is the union of Cartesian product and tensor product, and also is a binary operation on graphs. This operation takes two graphs and produces a new graph. In this paper, we will study the strong product on weighted graphs. The key to study the relationship between the spectrum of two original weighted graphs and that of their strong product graph is to provide a reasonable weight function to the weighted strong product graph. We introduce a definition of the weight function to the strong product graph $G\boxtimes H$ , where $G=(X,a)$ and $H=(Y,b)$ are two connected weighted graphs. And we derive an expression for the spectrum of $G\boxtimes H$ by using the spectrums of the weighted graph $G$ and $H$ . In this paper, we will also study the Ricci curvature of two adjacent points for the strong product. We prove that the Ricci curvature for strong product of two regular graphs with simple weight is bounded by the Ricci curvature of $G$ and $H$ .

Keywords

• Strong product
• spectrum
• Ricci curvature