In this paper, we define new graph operations F-composition F (G)[H], where F (G) be one of the symbols S(G),M(G),Q(G),T(G),Λ(G),Λ[G],D2(G),D2[G]. Further, we give some results for the Wiener indices of the these graph operations.

The Wiener index of a graph [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text] where the sum is taken through all unordered pairs of vertices of [Formula: see text] and [Formula: see text] is distance between two vertices [Formula: see text] and [Formula: see text] of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be two graphs. For a graph [Formula: see text], let [Formula: see text] be a copy of [Formula: see text] and [Formula: see text]. The [Formula: see text]-sum [Formula: see text] is a graph with the set of vertices [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] of [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text], where [Formula: see text] be one of the shadowgraph [Formula: see text] or closed shadowgraph [Formula: see text]. In this paper, we reported the Wiener index of these graphs.

is that graph whose vertex set is

Let $G = (V, E)$ be a graph. The \textit{Gallai total graph} $\Gamma_T(G)$ of $G$ is the graph, where $V(\Gamma_T(G))=V \cup E$ and $uv \in E(\Gamma_T(G))$ if and only if \begin{itemize} \item[$(i)$] $u$ and $v$ are adjacent vertices in $G$, or \item[$(ii)$] $u$ is incident to $v$ or $v$ is incident to $u$ in $G$, or \item[$(iii)$] $u$ and $v$ are adjacent edges in $G$ which do not span a triangle in $G$. \end{itemize}   The \textit{anti-Gallai total graph} $\Delta_T(G)$ of $G$ is the graph, where $V(\Delta_T(G))=V \cup E$ and $uv \in E(\Delta_T(G))$ if and only if \begin{itemize} \item[$(i)$] $u$ and $v$ are adjacent vertices in $G$, or \item[$(ii)$] $u$ is incident to $v$ or $v$ is incident to $u$ in $G$, or \item[$(iii)$] $u$ and $v$ are adjacent edges in $G$ and lie on a same triangle in $G$. \end{itemize}   In this paper, we discuss Eulerian and Hamiltonian properties of Gallai and anti-Gallai total graphs.DOI : http://dx.doi.org/10.22342/jims.21.2.230.105-116

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