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Subspace-based subspace sum graph on vector spaces
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DOI：10.1007/s00500-021-06006-7.png)
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Abstract
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Suppose that V is a n-dimensional vector space and W is its fixed k-dimensional subspace such that n - k = 1. In the present article, we initiate the study of a new graph structure subspace-based subspace sum graph G(W)(V), where the vertex set V(G(W)(V)) is the collection of all subspaces W of V such that W + W not equal V and W not subset of W, i.e., V(G(W)(V)) = {W subset of V vertical bar W + W not equal V, W not subset of W} and any two distinct vertices W-1 and W-1 of G(W)(V) are adjacent if and only if W-1 + W-2 + W = V. The girth, diameter, domination number, clique number, chromatic number and maximal independent sets of G(W)(V) have been studied. It is proved that two subspace-based subspace sum graphs G(W1)(V) and G(W2)(V) are isomorphic if and only if W-1 and W-2 are isomorphic. Further, we show that G(W)(V) is weakly perfect and characterize all vector spaces V and subspaces W of V for which G(W)(V) is perfect. Finally, in case of finite field degree of a given vertex, order and size of G(W)(V) have also been obtained.
Keywords:
Diameter
Connected graph
Domination
Subspace
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