Let $G$ be a finite group with the identity $e$. The subgroup intersection graph $\Gamma_{SI}(G)$ of $G$ is the graph with vertex set $V(\Gamma_{SI}(G)) = G-e$ and two distinct vertices $x$ and $y$ are adjacent in $\Gamma_{SI}(G)$ if and only if $|\left\langle x\right\rangle \cap\left\langle y\right\rangle|>1$, where $\left\langle x\right\rangle $ is the cyclic subgroup of $G$ generated by $x\in G$. In this paper, we obtain a lower bound for the independence number of subgroup intersection graph. We characterize certain classes of subgroup intersection graphs corresponding to finite abelian groups. Finally, we characterize groups whose automorphism group is the same as that of its subgroup intersection graph.
A. Abdollahi, S. Akbari and H. R. Maimani (2006). Non-commuting graph of a group. J. Algebra. 298, 468-492 P. J. Cameron, and S. Ghosh (2011). The power graph of a finite group. Discrete Math.. 311 (13), 1220-1222 G. Chartrand and P. Zhang (2006). Introduction to Graph Theory. Tata McGraw-Hill, New Delhi. I. Chakrabarty, S. Ghosh and M. K. Sen (2009). Undirected power graphs of semi groups. Semigroup Forum. 78, 410-426 Joseph A. Gallian (1999). Contemporary Abstract Algebra. Narosa Publishing House, New Delhi. A. V. Kelarev and S. J. Quinn (2002). Directed graph and Combinatorial properties of semi groups. J. Algebra. 251, 16-26 S. Lakshmivarahan, Jung-Sing Jwo and S. K. Dhall (1993). Symmetry in interconnection networks based on Cayley graphs of permutation groups: A survey. Parallel Comput.. 19, 361-407 T. Tamizh Chelvam and M. Sattanathan (2011). Subgroup intersection graph of a group. J. Adv. Research in Pure Math. (doi:10.5373/ jarpm .594.100910, ISSN:1943-2380).. 3 (4), 44-49