scholarly journals The Tight Upper Bound for the Number of Matchings of Tricyclic Graphs

10.37236/2165 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Ardeshir Dolati ◽  
Somayyeh Golalizadeh
Keyword(s):  
Upper Bound ◽  
Fibonacci Number ◽  
Tricyclic Graph ◽  
Tricyclic Graphs

In this paper, we determine the tight upper bound for the number of matchings of connected $n$-vertex tricyclic graphs. We show that this bound is $13 f_{n-4} + 16f_{n-5}$, where $f_n$ be the  $n$th Fibonacci number. We also  characterize the $n$-vertex simple connected tricyclic graph for which the bound is best  possible.A corrigendum was added to this paper on Jun 17, 2015. 

scholarly journals Bounds on F-index of tricyclic graphs with fixed pendant vertices

2020 ◽  
Vol 18 (1) ◽  
pp. 150-161
Author(s):  
Sana Akram ◽  
Muhammad Javaid ◽  
Muhammad Jamal
Keyword(s):  
Electron Energy ◽  
Connected Graph ◽  
Extremal Graphs ◽  
Complete Class ◽  
Zagreb Indices ◽  
Tricyclic Graph ◽  
Tricyclic Graphs

Abstract The F-index F(G) of a graph G is obtained by the sum of cubes of the degrees of all the vertices in G. It is defined in the same paper of 1972 where the first and second Zagreb indices are introduced to study the structure-dependency of total π-electron energy. Recently, Furtula and Gutman [J. Math. Chem. 53 (2015), no. 4, 1184–1190] reinvestigated F-index and proved its various properties. A connected graph with order n and size m, such that m = n + 2, is called a tricyclic graph. In this paper, we characterize the extremal graphs and prove the ordering among the different subfamilies of graphs with respect to F-index in $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \end{array}$, where $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \end{array}$ is a complete class of tricyclic graphs with three, four, six and seven cycles, such that each graph has α ≥ 1 pendant vertices and n ≥ 16 + α order. Mainly, we prove the bounds (lower and upper) of F(G), i.e $$\begin{array}{} \displaystyle 8n+12\alpha +76\leq F(G)\leq 8(n-1)-7\alpha + (\alpha+6)^3 ~\mbox{for each}~ G\in {\it\Omega}^{\alpha}_n. \end{array}$$

scholarly journals On numbers n dividing the nth term of a Lucas sequence

2017 ◽  
Vol 13 (03) ◽  
pp. 725-734 ◽  
Author(s):  
Carlo Sanna
Keyword(s):  
Upper Bound ◽  
Fibonacci Number ◽  
Lucas Sequence ◽  
Positive Integers

We prove that if [Formula: see text] is a Lucas sequence satisfying some mild hypotheses, then the number of positive integers [Formula: see text] does not exceed [Formula: see text] and such that [Formula: see text] divides [Formula: see text] is less than [Formula: see text] as [Formula: see text]. This generalizes a result of Luca and Tron about the positive integers [Formula: see text] dividing the [Formula: see text]th Fibonacci number, and improve a previous upper bound due to Alba González, Luca, Pomerance and Shparlinski.

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