Extremal multiplicative Zagreb indices among trees with given distance k-domination number

2020 ◽  
pp. 2150087
Author(s):  
Fazal Hayat
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Domination Number ◽  
Vertex Degree ◽  
Index Formula ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Sharp Lower Bound ◽  
Distance Formula

The first multiplicative Zagreb index [Formula: see text] of a graph [Formula: see text] is the product of the square of every vertex degree, while the second multiplicative Zagreb index [Formula: see text] is the product of the products of degrees of pairs of adjacent vertices. In this paper, we give sharp lower bound for [Formula: see text] and upper bound for [Formula: see text] of trees with given distance [Formula: see text]-domination number, and characterize those trees attaining the bounds.

scholarly journals ON THE PROPERTIES OF THE CAYLEY GRAPH OF RICHARD THOMPSON'S GROUP F

2004 ◽  
Vol 14 (05n06) ◽  
pp. 677-702 ◽  
Author(s):  
V. S. GUBA
Keyword(s):  
Growth Rate ◽  
Lower Bound ◽  
Cayley Graph ◽  
Upper Bound ◽  
Growth Function ◽  
Vertex Degree ◽  
General Growth ◽  
Thompson’S Group ◽  
Average Vertex ◽  
Thompson's Group F

We study some properties of the Cayley graph of R. Thompson's group F in generators x0, x1. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2-generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is also equivalent to the existence of a doubling function on the Cayley graph. This means there exists a mapping from the set of vertices into itself such that for some constant K>0, each vertex moves by a distance at most K and each vertex has at least two preimages. We show that the density of the Cayley graph of a 2-generated group does not exceed 3 if and only if the group satisfies the above condition with K=1. Besides, we give a very easy formula to find the length (norm) of a given element of F in generators x0, x1. This simplifies the algorithm by Fordham. The length formula may be useful for finding the general growth function of F in generators x0, x1 and the growth rate of this function. In this paper, we show that the growth rate of F has a lower bound of [Formula: see text].

THE DOMINATION NUMBER OF STRONG PRODUCT OF DIRECTED CYCLES

2014 ◽  
Vol 06 (02) ◽  
pp. 1450021
Author(s):  
HUIPING CAI ◽  
JUAN LIU ◽  
LINGZHI QIAN
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Domination Number ◽  
Strong Product ◽  
Directed Cycles

Let γ(D) denote the domination number of a digraph D and let Cm ⊗ Cn denote the strong product of Cm and Cn, the directed cycles of length m, n ≥ 2. In this paper, we determine the exact values [Formula: see text] Furthermore, we give a lower bound and an upper bound of γ(Cm1 ⊗ Cm2 ⊗ ⋯ ⊗ Cmn) and obtain that [Formula: see text] when at least n-2 integers of {m1, m2, …, mn} are even (because of the isomorphism, we assume that m3, m4, …, mn are even).

scholarly journals Total and Double Total Domination Number on Hexagonal Grid

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1110
Author(s):  
Antoaneta Klobučar ◽  
Ana Klobučar
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Hexagonal Grid

In this paper, we determine the upper and lower bound for the total domination number and exact values and the upper bound for the double-total domination number on hexagonal grid H m , n with m hexagons in a row and n hexagons in a column. Further, we explore the ratio between the total domination number and the number of vertices of H m , n when m and n tend to infinity.

Supremum and infimum of subharmonic functions of order between 1 and 2

2011 ◽  
Vol 54 (3) ◽  
pp. 685-693
Author(s):  
P. C. Fenton
Keyword(s):  
Lower Bound ◽  
Convex Function ◽  
Upper Bound ◽  
Counting Function ◽  
Subharmonic Functions ◽  
Image Position ◽  
Sharp Lower Bound

AbstractFor functions u, subharmonic in the plane, letand let N(r,u) be the integrated counting function. Suppose that $\mathcal{N}\colon[0,\infty)\rightarrow\mathbb{R}$ is a non-negative non-decreasing convex function of log r for which $\mathcal{N}(r)=0$ for all small r and $\limsup_{r\to\infty}\log\mathcal{N}(r)/\4\log r=\rho$, where 1 < ρ < 2, and defineA sharp upper bound is obtained for $\liminf_{r\to\infty}\mathcal{B}(r,\mathcal{N})/\mathcal{N}(r)$ and a sharp lower bound is obtained for $\limsup_{r\to\infty}\mathcal{A}(r,\mathcal{N})/\mathcal{N}(r)$.

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.

Lower Bounds for Matrices, II

1991 ◽  
Vol 44 (1) ◽  
pp. 54-74 ◽  
Author(s):  
Grahame Bennett
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Monotonicity Property ◽  
Hausdorff Matrix ◽  
Increasing Function ◽  
Sharp Lower Bound

AbstractOur main result is the following monotonicity property for moment sequences μ. Let p be fixed, 1 ≤ p < ∞: thenis an increasing function of r(r = 1,2,…). From this we derive a sharp lower bound for an arbitrary Hausdorff matrix acting on ℓp.The corresponding upper bound problem was solved by Hardy.

scholarly journals The Bondage Number of Random Graphs

10.37236/5180 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Dieter Mitsche ◽  
Xavier Pérez-Giménez ◽  
Paweł Prałat
Keyword(s):  
Lower Bound ◽  
Random Graph ◽  
Random Graphs ◽  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Side Product ◽  
Bondage Number ◽  
Concentration Result

A dominating set of a graph is a subset $D$ of its vertices such that every vertex not in $D$ is adjacent to at least one member of $D$. The domination number of a graph $G$ is the number of vertices in a smallest dominating set of $G$. The bondage number of a nonempty graph $G$ is the size of a smallest set of edges whose removal from $G$ results in a graph with domination number greater than the domination number of $G$. In this note, we study the bondage number of the binomial random graph $G(n,p)$. We obtain a lower bound that matches the order of the trivial upper bound. As a side product, we give a one-point concentration result for the domination number of $G(n,p)$ under certain restrictions.

scholarly journals Bounds and extremal graphs of second reformulated Zagreb index for graphs with cyclomatic number at most three

2021 ◽  
Vol 49 (1) ◽  
Author(s):  
Abhay Rajpoot ◽  
◽  
Lavanya Selvaganesh ◽  
Keyword(s):  
Lower Bounds ◽  
Topological Indices ◽  
Simple Approach ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Cyclomatic Number ◽  
Tricyclic Graphs

Miliˇcevi´c et al., in 2004, introduced topological indices known as Reformulated Zagreb indices, where they modified Zagreb indices using the edge-degree instead of vertex degree. In this paper, we present a simple approach to find the upper and lower bounds of the second reformulated Zagreb index, EM2(G), by using six graph operations/transformations. We prove that these operations significantly alter the value of reformulated Zagreb index. We apply these transformations and identify those graphs with cyclomatic number at most 3, namely trees, unicyclic, bicyclic and tricyclic graphs, which attain the upper and lower bounds of second reformulated Zagreb index for graphs.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

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