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\iteman{io-port 05943495}
\itemau{Reddy, P.Siva Kota; Prashanth, B.; Permi, Kavita S.}
\itemti{A note on antipodal signed graphs.}
\itemso{Int. J. Math. Comb. 1, 107-112 (2011).}
\itemab
Summary: A Smarandachely $k$-signed graph (Smarandachely $k$-marked graph) is an ordered pair $S= (G,\sigma)$ $(S= (G,\mu))$ where $G= (V,E)$ is a graph called underlying graph of $S$ and $\sigma: E\to (\overline e_1,\overline e_2,\dots,\overline e_k)$ ($\mu: V \to(\overline e_1,\overline e_2,\dots,\overline e_k)$) is a function, where each $\overline e_i\in\{+,-\}$. Particularly, a Smarandachely 1-signed graph or Smarandachely 1-marked graph is called abbreviated a signed graph or a marked graph. {\it R. Singleton} [Am. Math. Mon. 75, 42--43 (1968; Zbl 0173.26303)] introduced the concept of the antipodal graph of a graph $G$, denoted by $A(G)$, is the graph on the same vertices as of $G$, two vertices being adjacent if the distance between them is equal to the diameter of $G$. Analogously, one can define the antipodal signed graph $A(S)$ of a signed graph $S= (G,\sigma)$ as a signed graph, $A(S)= (A(G),\sigma')$, where $A(G)$ is the underlying graph of $A(S)$, and for any edge $e= uv$ in $A(S)$, $\sigma'(e)= \mu(u)\mu(v)$, where for any $v\in V$, $\mu(v)=\prod_{v\in N(v)} \sigma(uv)$. It is shown that for any signed graph $S$, its $A(S)$ is balanced and we offer a structural characterization of antipodal signed graphs. Further, we characterize signed graphs $S$ for which $S\sim A(S)$ and $\overline S\sim A(S)$ where $\sigma$ denotes switching equivalence and $A(S)$ and $\overline S$ are denotes the antipodal signed graph and complementary signed graph of $S$, respectively.
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\itemcc{}
\itemut{Smarandachely $k$-signed graphs; Smarandachely $k$-marked graphs; signed graphs; marked graphs; balance; switching; antipodal signed graphs; complement; negation}
\itemli{}
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