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.
A $signed graph$ (or $sigraph$ in short) is an ordered pair $S = (S^u, \sigma)$, where $S^u$ is a graph $G = (V, E)$ and $\sigma : E\rightarrow \{+,-\}$ is a function from the edge set $E$ of $S^u$ into the set $\{+, -\}$. For a positive integer $n > 1$, the unitary Cayley graph $X_n$ is the graph whose vertex set is $Z_n$, the integers modulo $n$ and if $U_n$ denotes set of all units of the ring $Z_n$, then two vertices $a, b$ are adjacent if and only if $a-b \in U_n$. For a positive integer $n > 1$, the unitary Cayley sigraph $\mathcal{S}_n = (\mathcal{S}^u_n, \sigma)$ is defined as the sigraph, where $\mathcal{S}^u_n$ is the unitary Cayley graph and for an edge $ab$ of $\mathcal{S}_n$, $$\sigma(ab) = \begin{cases} + & \text{if } a \in U_n \text{ or } b \in U_n,\\ - & \text{otherwise.} \end{cases}$$ In this paper, we have obtained a characterization of balanced unitary Cayley sigraphs. Further, we have established a characterization of canonically consistent unitary Cayley sigraphs $\mathcal{S}_n$, where $n$ has at most two distinct odd prime factors.
A signed graph (sigraph) is an ordered pair S ¼ ðS u ; rÞ; where S u is a graph G ¼ ðV; EÞ and r : E ! fþ; Àg is a function from the edge set E of S u into the set {?, -}. The canonical marking on S is defined as: for each vertex v 2 VðSÞ; l r ðvÞ ¼ Ythe value of marking of v is negative. Let S be canonically marked, then a cycle Z in S is said to be canonically consistent if it contains an even number of negative vertices. If every cycle in S is canonically consistent, then S is called canonically consistent. In this paper, we characterize canonically consistent semi-total line sigraphs.
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