Connected geodesic number of a fuzzy graph
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In this paper, the concept of connected geodesic number, $ gn_{c}(G) $, of a fuzzy graph $ G $ is introduced and its limiting bounds are identified. It is proved that all extreme nodes of $ G $ and all cut-nodes of the underlying crisp graph $ G^{*} $ belong to every connected geodesic cover of $ G $. The connected geodesic number of complete fuzzy graphs, fuzzy cycles, fuzzy trees and of complete bipartite fuzzy graphs are obtained. It is proved that for any pair $ k,n $ of integers with $ 3\leq k\leq n $, there exists a connected fuzzy graph $ G:(V,\sigma,\mu) $ on $ n $ nodes such that $ gn_{c}(G)=k $. Also, for any positive integers $ 2 \leq a < b \leq c $, it is proved that there exists a connected fuzzy graph $ G:(V,\sigma,\mu) $ such that the geodesic number $ gn(G)=a $ and the connected geodesic number $ gn_{c}(G)=b $.
Abstract
In this paper, the concept of connected geodesic number, $ gn_{c}(G) $, of a fuzzy graph $ G $ is introduced and its limiting bounds are identified. It is proved that all extreme nodes of $ G $ and all cut-nodes of the underlying crisp graph $ G^{*} $ belong to every connected geodesic cover of $ G $. The connected geodesic number of complete fuzzy graphs, fuzzy cycles, fuzzy trees and of complete bipartite fuzzy graphs are obtained. It is proved that for any pair $ k,n $ of integers with $ 3\leq k\leq n $, there exists a connected fuzzy graph $ G:(V,\sigma,\mu) $ on $ n $ nodes such that $ gn_{c}(G)=k $. Also, for any positive integers $ 2 \leq a < b \leq c $, it is proved that there exists a connected fuzzy graph $ G:(V,\sigma,\mu) $ such that the geodesic number $ gn(G)=a $ and the connected geodesic number $ gn_{c}(G)=b $.
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