On m-Bonacci-Sum Graphs
14 Feb 2019-pp 65-76
TL;DR: The notion of m-bonacci-sum graphs denoted by \(G_{m,n}\) for positive integers m, n are introduced and it is shown that this graph is bipartite and for \(n\ge 2^{m-2}\), \(\{1,2,\ldots ,n\) has exactly \((m-1)\) components.
Abstract: We introduce the notion of m-bonacci-sum graphs denoted by \(G_{m,n}\) for positive integers m, n. The vertices of \(G_{m,n}\) are \(1,2,\ldots ,n\) and any two vertices are adjacent if and only if their sum is an m-bonacci number. We show that \(G_{m,n}\) is bipartite and for \(n\ge 2^{m-2}\), \(G_{m,n}\) has exactly \((m-1)\) components. We also find the values of n such that \(G_{m,n}\) contains cycles as subgraphs. We also use this graph to partition the set \(\{1,2,\ldots ,n\}\) into \(m-1\) subsets such that each subset is ordered in such a way that sum of any 2 consecutive terms is an m-bonacci number.
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TL;DR: A graph G on n edges is m-bonacci graceful if the vertices can be labeled with distinct integers from the set { 0, 1, 2, …, Z n, m...
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30 citations
TL;DR: In this paper, two new summation formulas for M-bonacci numbers are given, which are generalizations of the two summation formula for Fibonacci numbers, and the formulas are shown to be equivalent to the two formulas for the M-Bonacci numbers.
Abstract: The note considers M-bonacci numbers, which are a generalization of Fibonacci numbers. Two new summation formulas for M-bonacci numbers are given. The formulas are generalizations of the two summation formulas for Fibonacci numbers.
9 citations
TL;DR: In this article, two summation formulas for sequences with M-bonacci property were derived and applied to the arithmetic series, mth g − gonal numbers and the Fibonacci series.
Abstract: The note introduces sequences having M-bonacci property. Two summation formulas for sequences with M-bonacci property are derived. The formulas are generalizations of corresponding summation formulas for both M-bonacci numbers and Fibonacci numbers that have appeared previously in the literature. Applications to the Arithmetic series, mth g − gonal numbers and the Fibonacci series are demonstrated.
8 citations