‎let $r$ be a commutative ring with non-zero identity. ‎we describe all $c_3$‎- ‎and $c_4$-free intersection graph of non-trivial ideals of $r$ as well as $c_n$-free intersection graph when $r$ is a reduced ring. ‎also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎finally, ‎we shall prove that almost all artin rings $r$ have hamiltonian intersection graphs. ...

We analyse an extremal question on the degrees of link graphs a finite regular graph, that is, subgraphs induced by non-trivial spheres. show if $G$ is $d$-regular and connected but not complete then some graph has minimum degree at most $\lfloor 2d/3\rfloor-1$, sufficiently large in terms $d$ d/2\rfloor-1$; both bounds are best possible. also give corresponding best-possible result for problem...