scholarly journals Total Graph Interpretation of the Numbers of the Fibonacci Type

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Urszula Bednarz ◽  
Iwona Włoch ◽  
Małgorzata Wołowiec-Musiał
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Total Graph ◽  
Graph Interpretation ◽  
Edge Colourings ◽  
Almost All ◽  
Edge Colouring

We give a total graph interpretation of the numbers of the Fibonacci type. This graph interpretation relates to an edge colouring by monochromatic paths in graphs. We will show that it works for almost all numbers of the Fibonacci type. Moreover, we give the lower bound and the upper bound for the number of all(A1,2A1)-edge colourings in trees.

scholarly journals A Note on Edge-Colourings Avoiding Rainbow $K_4$ and Monochromatic $K_m$

10.37236/257 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Veselin Jungić ◽  
Tomáš Kaiser ◽  
Daniel Král'
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Projective Planes ◽  
Ramsey Number ◽  
Complete Subgraph ◽  
Finite Projective Planes ◽  
Edge Colourings ◽  
Edge Colouring

We study the mixed Ramsey number $maxR(n,{K_m},{K_r})$, defined as the maximum number of colours in an edge-colouring of the complete graph $K_n$, such that $K_n$ has no monochromatic complete subgraph on $m$ vertices and no rainbow complete subgraph on $r$ vertices. Improving an upper bound of Axenovich and Iverson, we show that $maxR(n,{K_m},{K_4}) \leq n^{3/2}\sqrt{2m}$ for all $m\geq 3$. Further, we discuss a possible way to improve their lower bound on $maxR(n,{K_4},{K_4})$ based on incidence graphs of finite projective planes.

scholarly journals On r-dynamic coloring of comb graphs

2021 ◽  
Vol 27 (2) ◽  
pp. 191-200
Author(s):  
K. Kalaiselvi ◽  
◽  
N. Mohanapriya ◽  
J. Vernold Vivin ◽  
◽  
...  
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Line Graph ◽  
Proper Coloring ◽  
Total Graph ◽  
Middle Graph ◽  
Different Color

An r-dynamic coloring of a graph G is a proper coloring of G such that every vertex in V(G) has neighbors in at least $\min\{d(v),r\}$ different color classes. The r-dynamic chromatic number of graph G denoted as $\chi_r (G)$, is the least k such that G has a coloring. In this paper we obtain the r-dynamic chromatic number of the central graph, middle graph, total graph, line graph, para-line graph and sub-division graph of the comb graph $P_n\odot K_1$ denoted by $C(P_n\odot K_1), M(P_n\odot K_1), T(P_n\odot K_1), L(P_n\odot K_1), P(P_n\odot K_1)$ and $S(P_n\odot K_1)$ respectively by finding the upper bound and lower bound for the r-dynamic chromatic number of the Comb graph.

scholarly journals Neighbour$\,$–$\,$Distinguishing Edge Colourings of Random Regular Graphs

10.37236/1103 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Catherine Greenhill ◽  
Andrzej Ruciński
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Edge Colourings ◽  
Almost All ◽  
Edge Colouring

A proper edge colouring of a graph is neighbour-distinguishing if for all pairs of adjacent vertices $v$, $w$ the set of colours appearing on the edges incident with $v$ is not equal to the set of colours appearing on the edges incident with $w$. Let ${\rm ndi}(G)$ be the least number of colours required for a proper neighbour-distinguishing edge colouring of $G$. We prove that for $d\geq 4$, a random $d$-regular graph $G$ on $n$ vertices asymptotically almost surely satisfies ${\rm ndi}(G)\leq \lceil 3d/2\rceil$. This verifies a conjecture of Zhang, Liu and Wang for almost all 4-regular graphs.

scholarly journals Iterative estimation of total π-electron energy

2005 ◽  
Vol 70 (10) ◽  
pp. 1193-1197 ◽  
Author(s):  
Lemi Türker ◽  
Ivan Gutman
Keyword(s):  
Lower Bound ◽  
Electron Energy ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Benzenoid Hydrocarbons ◽  
Total Electron ◽  
Iterative Estimation ◽  
Total Electron Energy ◽  
Almost All

In this work, the lower and upper bounds for total ?-electron energy (E) was studied. A method is presented, by means of which, starting with a lower bound EL and an upper bound EU for E, a sequence of auxiliary quantities E0 E1, E2,? is computed, such that E0 = EL, E0 < E1 < E2 < ?, and E = EU. Therefore, an integer k exists, such that Ek E < Ek+1. If the estimates EL and EU are of the McClelland type, then k is called the McClelland number. For almost all benzenoid hydrocarbons, k = 3.

A (5,5)-Colouring of Kn with Few Colours

2018 ◽  
Vol 27 (6) ◽  
pp. 892-912
Author(s):  
ALEX CAMERON ◽  
EMILY HEATH
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Minimum Number ◽  
Edge Colouring

For fixed integers p and q, let f(n,p,q) denote the minimum number of colours needed to colour all of the edges of the complete graph Kn such that no clique of p vertices spans fewer than q distinct colours. Any edge-colouring with this property is known as a (p,q)-colouring. We construct an explicit (5,5)-colouring that shows that f(n,5,5) ≤ n1/3 + o(1) as n → ∞. This improves upon the best known probabilistic upper bound of O(n1/2) given by Erdős and Gyárfás, and comes close to matching the best known lower bound Ω(n1/3).

Linear Equations with Small Prime and Almost Prime Solutions

2008 ◽  
Vol 51 (3) ◽  
pp. 399-405
Author(s):  
Xianmeng Meng
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Linear Equations ◽  
Prime Factors ◽  
Almost All ◽  
Sharp Lower Bound ◽  
Almost Prime

AbstractLet b1, b2 be any integers such that gcd(b1, b2) = 1 and c1|b1| < |b2| ≤ c2|b1|, where c1, c2 are any given positive constants. Let n be any integer satisfying gcd(n, bi) = 1, i = 1, 2. Let Pk denote any integer with no more than k prime factors, counted according to multiplicity. In this paper, for almost all b2, we prove (i) a sharp lower bound for n such that the equation b1p + b2m = n is solvable in prime p and almost prime m = Pk, k ≥ 3 whenever both bi are positive, and (ii) a sharp upper bound for the least solutions p, m of the above equation whenever bi are not of the same sign, where p is a prime and m = Pk, k ≥ 3.

scholarly journals Multiple closed orbits for N-body-type problems

1998 ◽  
Vol 58 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Shiqing Zhang
Keyword(s):  
Lower Bound ◽  
Periodic Solutions ◽  
Upper Bound ◽  
Critical Value ◽  
Body Type ◽  
Fixed Energy ◽  
Closed Orbits

Using the equivariant Ljusternik-Schnirelmann theory and the estimate of the upper bound of the critical value and lower bound for the collision solutions, we obtain some new results in the large concerning multiple geometrically distinct periodic solutions of fixed energy for a class of planar N-body type problems.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

Isolated minima of the product of n linear forms

1953 ◽  
Vol 49 (1) ◽  
pp. 59-62 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Linear Forms ◽  
Image Position

Letbe n linear forms with real coefficients and determinant Δ = ∥ aij∥ ≠ 0; and denote by M(X) the lower bound of | X1X2 … Xn| over all integer sets (u) ≠ (0). It is well known that γn, the upper bound of M(X)/|Δ| over all sets of forms Xi, is finite, and the value of γn has been determined when n = 2 and n = 3.

scholarly journals The Algebraic Degree of Phase-Type Distributions

2010 ◽  
Vol 47 (03) ◽  
pp. 611-629
Author(s):  
Mark Fackrell ◽  
Qi-Ming He ◽  
Peter Taylor ◽  
Hanqin Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Polynomial Algorithm ◽  
Algebraic Degree ◽  
Nonnegative Matrices ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

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