The profile of the Cartesian product of graphs Original Research Article

Given a graph image, a proper labeling image of image is a one-to-one function from image onto image. For a proper labeling image of image, the profile width image of a vertex image is the minimum value of image, where image belongs to the closed neighborhood of image. The profile of a proper labeling image of image, denoted by image, is the sum of all the image, where image. The profile of image is the minimum value of image, where image runs over all proper labeling of image. In this paper, we show that if the vertices of a graph image can be ordered to satisfy a special neighborhood property, then so can the graph image. This can be used to determine the profile of image and image.