We study the connectivity of proper power graphs of some family of finite groups including nilpotent groups, groups with a nontrivial partition, and symmetric and alternating groups. Also, for such a group, the corresponding proper power graph has diameter at most 26 whenever it is connected.
We have classified, upto isoclinism, certain groups with a given central factor. As an application, we classify, upto isoclinism, groups having at the most nine element centralizers. Among other results of independent interest, we have classified, upto isoclinism, groups having a central factor of order p 3 , p a prime. All these improves some previous results. 2020 Mathematics Subject Classification. 20D60, 20D99.
A lattice path L in Z d of length k with steps in a given set S ⊆ Z d , or S-path for short, is a sequence ν 1 , ν 2 , . . . , ν k ∈ Z d such that the steps ν i − ν i−1 lie in S for all i = 2, . . . , k. Let T m,n be the m × n table in the first area of xy-axis and put S = {(1, 0), (1, 1), (1, −1)}. Accordingly, let I m (n) denote the number of S-paths starting from the first column and ending at the last column of T . We will study the numbers I m (n) and give explicit formulas for special values of m and n. As a result, we prove a conjecture of Alexander R. Povolotsky. We conclude the paper with some applications to Fibonacci and Pell-Lucas numbers and posing an open problem.
Abstract. All nite simple self -distance graphs with no square, diamond, or triangles with a common vertex as subgraph are determined. Utilizing these results, it is shown that there is no cubic self -distance graph.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.