Automorphism Group and Distinguishing Number of Some Shadow Graphs and Some Split Graphs

Abstract

The proposed study asserts the automorphism group and distinguishing number of some shadow graphs along with some split graphs. Automorphism group of path graph $P_{n}$, cycle graph $C_{n}$ and star graph $K_{1,n}$ are well-known groups and also their distinguishing number are well-known. It is full of zest to know what would be the automorphism group of shadow graphs and split graphs of that graphs whose automorphism group are known. Also, it would be interesting to determine the distinguishing number of shadow graphs and split graphs of that graphs whose distinguishing numbers are known. In this paper, shadow graph as well as split graph of path graph $P_{n}$, cycle graph $C_{n}$ and star graph $K_{1,n}$ have been taken into the account to investigate their the automorphism groups and distinguishing numbers. The results are illustrated with the help of examples and the applicability of the proposed theory, related to some shadow graphs and some split graphs, is elaborated successfully.