‎The Hosoya index and the Merrifield-Simmons index are two types of‎ ‎graph invariants used in mathematical chemistry‎. ‎In this paper‎, ‎we give some formulas to compute these indices for some classes‎ ‎of corona product and link of two graphs‎. ‎Furthermore‎, ‎we obtain‎ ‎exact formulas of Hosoya and Merrifield-Simmons indices for the‎ ‎set of bicyclic graphs‎, ‎caterpillars and dual star‎.

Let G be a nite group with the identity e. The subgroup intersection graph SI(G) of G is the graph with vertex set V ( SI(G)) = G e and two distinct vertices x and y are adjacent in SI(G) if and only ifjhxi\hyij > 1, wherehxi is the cyclic subgroup of G generated by x2 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 nite abelian groups. Finally, we characterize groups whose automorphism group is the same as that of its subgroup intersection graph.

‎For a given graph $G=(V,E)$‎, ‎let $mathscr L(G)={L(v)‎ : ‎vin V}$ be a prescribed list assignment‎. ‎$G$ is $mathscr L$-$L(2,1)$-colorable if there exists a vertex labeling $f$ of $G$ such that $f(v)in L(v)$ for all $v in V$; $|f(u)-f(v)|geq 2$ if $d_G(u,v) = 1$; and $|f(u)-f(v)| geq 1$ if $d_G(u,v)=2$‎. ‎If $G$ is $mathscr L$-$L(2,1)$-colorable for every list assignment $mathscr L$ with $|L(v)|geq k$ for all $vin V$‎, ‎then $G$ is said to be $k$-$L(2,1)$-choosable‎. ‎In this paper‎, ‎we prove all cycles are $5$-$L(2,1)$-choosable‎.

Let $G$ be a connected graph with vertex set $V(G)$‎. ‎The‎ ‎degree resistance distance of $G$ is defined as $D_R(G) = sum_{{u‎, ‎v} subseteq V(G)} [d(u)+d(v)] R(u,v)$‎, ‎where $d(u)$ is the degree‎ ‎of vertex $u$‎, ‎and $R(u,v)$ denotes the resistance distance between‎ ‎$u$ and $v$‎. ‎In this paper‎, ‎we characterize $n$-vertex unicyclic‎ ‎graphs having minimum and second minimum degree resistance‎ ‎distance‎. ‎

A dominating setD V of a graphG = (V;E) is said to be a connected cototal dominating set ifhDi is connected andhV Di6= ?, contains no isolated vertices. A connected cototal dominating set is said to be minimal if no proper subset of D is connected cototal dominating set. The connected cototal domination number ccl(G) of G is the minimum cardinality of a minimal connected cototal dominating set of G. In this paper, we begin an investigation of connected cototal domination number and obtain some interesting results.