Let $G=(V,E)$ be a connected graph with the vertex-set $V$ and  the edge-set $E$.    The subdivision graph $S(G)$ of the graph $G$ is obtained from $G$ by adding a vertex in the middle of every edge of $G$.  In this paper, we investigate some properties of the graphs  $S(G)$ and $L(S(G))$, where $L(S(G))$ is the line graph of $S(G)$. We will see that $S(G)$ and  $L(S(G))$  inherit some  properties of $G$.    For instance, we show that if $G \ncong C_n$, then $Aut(G) \cong Aut(L(S(G)))$ (as abstract groups), where $C_n$ is the cycle of order $n$.

A Roman dominating function on a graph G= (V, E) is a function f: V®{0, 1, 2} satisfying the condition that every vertex v for which f (v) =0 is adjacent to at least one vertex u for which f (u) =2. The weight of a Roman dominating function is the value w (f) = S vÎV f (v). The Roman domination number of a graph G, denoted by yR (G), equals the minimum weight of a Roman dominating function on G. The Roman game domination subdivision number of a graph G is defined by the following game. Two players D and A, D playing first, alternately mark or subdivide an edge of G which is not yet marked nor subdivided. The game ends when all the edges of G are marked or subdivided and results in a new graph G’. The purpose of D is to minimize the Roman domination number yR (G’) of G’ while A tries to maximize it. If both A and D play according to their optimal strategies, R (G0) is well defined. We call this number the Roman game domination subdivision number of G and denote it by yRgs (G). In this paper we initiate the study of the Roman game domination subdivision number of a graph and present sharp bounds on the Roman game domination subdivision number of a tree.

Expressions for the Zagreb indices and coindices of the total graph, semitotal point graph and of semi-total line graph of subdivision graphs in terms of the parameters of the parent graph are obtained, thus generalizing earlier existing results.

The harmonic index of a graph G is defined as the sum of the weights (Formula)of all edges uv of G, where deg G (u) denotes the degree of a vertex u in G. In this paper, we study the harmonic index of subdivision graphs, t -subdivision graphs and also, S -sum and St -sum of graphs.

In this paper, we obtain M-polynomial of some graph operations and cy-cle related graphs. As an application, we compute M-polynomial of some nanostruc-tures viz., TUC4C8[p; q] nanotube, TUC4C8[p; q] nanotorus, line graph of subdivision graph of TUC4C8[p; q] nanotube and TUC4C8[p; q] nanotorus, V-tetracenic nanotube and V-tetracenic nanotorus. Further, we derive some degree based topological indices from the obtained polynomials.

For any kÎN, the k -subdivision of a graph G is a simple graph G 1/k , which is constructed by replacing each edge of G with a path of length k. In [Moharram N. Iradmusa, On colorings of graph fractional powers, Discrete Math., (310) 2010, No.10-11, 1551-1556] the m th power of the n-subdivision of G has been introduced as a fractional power of G, denoted by G m/n. In this regard, we investigate domination number and independent domination number of fractional powers of graphs.

A Roman dominating function on a graph G is a function f: V (G)! f0; 1; 2g satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. A total Roman dominating function is a Roman dominating function with the additional property that the subgraph of G induced by the set of all vertices of positive weight has no isolated vertices. The weight of a total Roman dominating function f is the value  u2V (G)f(u). The total Roman domination number of G, tR(G), is the minimum weight of a total Roman dominating function on G. The total Roman domination subdivision number sd tR (G) of a graph G is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total Roman domination number. In this paper, we initiate the study of total Roman domination subdivision number in graphs and we present sharp bounds for this parameter.

The paired domination subdivision number of a graph G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the paired domination number of G. In this note, we show that the problem of computing the paired domination subdivision number is NP-hard for bipartite graphs.