In this paper we introduce k-isolate dominating set and minimal k-isolate dominating set and it is defined as follows: A dominating set S of a graph G is said to be a k-isolate dominating set if S has at least k-isolated vertices. The k-isolate dominating set S is said to be a minimal k-isolate dominating set if any proper subset of S is not an isolate dominating set. The minimum and the maximum cardinality of a minimal k-isolate dominating set of G are called k-isolate domination number denoted by γ ki (G) and the upper k-isolate domination number by Γ ki (G) respectively. Also k-isolate domination number for some simple graphs such as path, cycle and wheel graph have been found.
In this paper, we have introduced a new generalization of Jacobsthal polynomial (bi-periodic Jacobsthal polynomial), have obtained Binet’s formula, generating function, well-known Cassini’s, Catalan ’s and d’Ocagne’s Identities and some more results related to this polynomial.
Let G be an arbitrary-group, where-groups are groups with number of centralizers & is any finite number. In this article, we have proved that the group of inner automorphisms of G is isomorphic to some other groups depending upon. Moreover if for some group , the group of inner automorphisms () has order 6 or 9 then will be 5-group and if for some group , the group of inner automorphisms () has order 4 then will be 4-group & conversely.
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