On Powers of Some Graph Operations

Document Type: Original Scientific Paper

Authors

1 Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, Cairo, Egypt

2 Department of Basic science, Faculty of Computers and Information, Fayoum University, Fayoum 63514, Egypt

Abstract

Let $G*H$ be the product $*$ of $G$ and $H$. In this paper we determine the rth power of the graph $G*H$ in terms of $G^r, H^r$ and $G^r*H^r$, when $*$ is the join, Cartesian, symmetric difference, disjunctive, composition, skew and corona product. Then we solve the equation $(G*H)^r=G^r*H^r$. We also compute the Wiener index and Wiener polarity index of the skew product.

Keywords

Main Subjects


1. R. Hammack, W. Imrich, S. Klav┼żar, Handbook of Product Graphs, CRC press, Boca Raton, FL, 2011.

2. F. Harary, Graph Theory, Addison-Wesley Publishing Co., Reading, Mass. -Menlo Park, Calif. -London 1969.

3. F. Harary, G. W. Wilcox, Boolean operations on graphs, Math. Scand. 26(1) (1967) 41–51.

4. J. Ma, Y. Shi, J. Yue, The Wiener polarity index of graph products, Ars Combin. 116 (2014) 235–244.

5. S. Moradi, A note on tensor product of graphs, Iran. J. Math. Sci. Inform. 7(1) (2012) 73–81.

6. I. Peterina, P. Z. Pleteršek, Wiener index of strong product of graphs, Opuscula Math. 38(1) (2018) 81–94.

7. M. A. Seoud, On square graphs, Proc. Pakistan Acad. Sci. 25(1) (1991) 35–42.

8. M. A. Seoud, Operations related to squaring of graphs, Proc. Pakistan Acad. Sci. 28(3) (1991) 303–309.

9. M. A. Seoud, On power graphs, Ain Shams Science Bulletin 29(A) (1992) 125–135.

10. Y. Shibata, Y. Kikuchi, Graph products based on the distance in graphs, IEICE Trans. Fundamentals E83-A(3) (2000) 459-464.

11. H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69(1) (1947) 17–20.

12. Y. N. Yeh, I. Gutman, On the sum of all distances in composite graphs, Discrete Math. 135(1-3) (1994) 359–365.