In the present note, we introduce and study distinct subspace distinct component graph \(\Gamma ^{\star }_{\mathcal {B}}(\mathbb {V})\) and prove that \(\Gamma ^{\star }_{\mathcal {B}}(\mathbb {V})\) is connected with diameter two. The clique number and chromatic number of \(\Gamma ^{\star }_{\mathcal {B}}(\mathbb {V})\) have been studied. Further, we show that \(\Gamma ^{\star }_{\mathcal {B}}(\mathbb {V})\) is perfect. It is shown that two distinct subspace distinct component graphs \(\Gamma ^{\star }_{\mathcal {B}_{1}}(\mathbb {V}_{1})\) , and \(\Gamma ^{\star }_{\mathcal {B}_{2}}(\mathbb {V}_{2})\) are isomorphic if and only if \(\mathbb {V}_{1}\) and \(\mathbb {V}_{2}\) are isomorphic. Finally, in case of finite field, Planarity of \(\Gamma ^{\star }_{\mathcal {B}}(\mathbb {V})\) have also been studied and proved that genus of \(\Gamma ^{\star }_{\mathcal {B}}(\mathbb {V})\ne 1,2,\ldots ,5\) .

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DSDC Graphs on Finite-Dimensional Vector Spaces

  • Mohit Kumar,
  • J. H. Asalool,
  • Nazia Parveen

摘要

In the present note, we introduce and study distinct subspace distinct component graph \(\Gamma ^{\star }_{\mathcal {B}}(\mathbb {V})\) and prove that \(\Gamma ^{\star }_{\mathcal {B}}(\mathbb {V})\) is connected with diameter two. The clique number and chromatic number of \(\Gamma ^{\star }_{\mathcal {B}}(\mathbb {V})\) have been studied. Further, we show that \(\Gamma ^{\star }_{\mathcal {B}}(\mathbb {V})\) is perfect. It is shown that two distinct subspace distinct component graphs \(\Gamma ^{\star }_{\mathcal {B}_{1}}(\mathbb {V}_{1})\) , and \(\Gamma ^{\star }_{\mathcal {B}_{2}}(\mathbb {V}_{2})\) are isomorphic if and only if \(\mathbb {V}_{1}\) and \(\mathbb {V}_{2}\) are isomorphic. Finally, in case of finite field, Planarity of \(\Gamma ^{\star }_{\mathcal {B}}(\mathbb {V})\) have also been studied and proved that genus of \(\Gamma ^{\star }_{\mathcal {B}}(\mathbb {V})\ne 1,2,\ldots ,5\) .