scholarly journals The Extremal Function and Colin de Verdière Graph Parameter

10.37236/7195 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Rose McCarty
Keyword(s):  
Differential Geometry ◽  
Upper Bound ◽  
Extremal Function ◽  
The Extremal Function ◽  
A Minor ◽  
Monotone Graph

The Colin de Verdière parameter $\mu(G)$ is a minor-monotone graph parameter with connections to differential geometry. We study the conjecture that for every integer $t$, if $G$ is a graph with at least $t$ vertices and $\mu(G) \leq t$, then $|E(G)| \leq t|V(G)|-\binom{t+1}{2}$. We observe a relation to the graph complement conjecture for the Colin de Verdière parameter and prove the conjectured edge upper bound for graphs $G$ such that either $\mu(G) \leq 7$, or $\mu(G) \geq |V(G)|-6$, or the complement of $G$ is chordal, or $G$ is chordal.

scholarly journals Bounds on Parameters of Minimally Nonlinear Patterns

10.37236/6735 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
P. A. CrowdMath
Keyword(s):  
Upper Bound ◽  
Extremal Function ◽  
Subgraph Isomorphic ◽  
Non Linear ◽  
Ordered Graphs ◽  
P Matrix ◽  
The Extremal Function

Let $ex(n, P)$ be the maximum possible number of ones in any 0-1 matrix of dimensions $n \times n$ that avoids $P$. Matrix $P$ is called minimally non-linear if $ex(n, P) \neq O(n)$ but $ex(n, P') = O(n)$ for every proper subpattern $P'$ of $P$. We prove that the ratio between the length and width of any minimally non-linear 0-1 matrix is at most $4$, and that a minimally non-linear 0-1 matrix with $k$ rows has at most $5k-3$ ones. We also obtain an upper bound on the number of minimally non-linear 0-1 matrices with $k$ rows.In addition, we prove corresponding bounds for minimally non-linear ordered graphs. The minimal non-linearity that we investigate for ordered graphs is for the extremal function $ex_{<}(n, G)$, which is the maximum possible number of edges in any ordered graph on $n$ vertices with no ordered subgraph isomorphic to $G$.

scholarly journals Sharper Bounds and Structural Results for Minimally Nonlinear 0-1 Matrices

10.37236/7801 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Jesse Geneson ◽  
Shen-Fu Tsai
Keyword(s):  
Upper Bound ◽  
Extremal Function ◽  
Nonlinear Matrix ◽  
The Extremal Function

The extremal function $ex(n, P)$ is the maximum possible number of ones in any 0-1 matrix with $n$ rows and $n$ columns that avoids $P$. A 0-1 matrix $P$ is called minimally nonlinear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every $P'$ that is contained in $P$ but not equal to $P$. Bounds on the number of ones and the number of columns in a minimally nonlinear 0-1 matrix with $k$ rows were found in (CrowdMath, 2018). In this paper, we improve the upper bound on the number of ones in a minimally nonlinear 0-1 matrix with $k$ rows from $5k-3$ to $4k-4$. As a corollary, this improves the upper bound on the number of columns in a minimally nonlinear 0-1 matrix with $k$ rows from $4k-2$ to $4k-4$. We also prove that there are not more than four ones in the top and bottom rows of a minimally nonlinear matrix and that there are not more than six ones in any other row of a minimally nonlinear matrix. Furthermore, we prove that if a minimally nonlinear 0-1 matrix has ones in the same row with exactly $d$ columns between them, then within these columns there are at most $2d-1$ rows above and $2d-1$ rows below with ones.

scholarly journals A majorant problem

1992 ◽  
Vol 15 (3) ◽  
pp. 441-447
Author(s):  
Ronen Peretz
Keyword(s):  
Unit Disc ◽  
Extremal Function ◽  
Fourier Coefficients ◽  
Positive Real Part ◽  
Complex Vector ◽  
Positive Real ◽  
Zero Sets ◽  
The Extremal Function

Letf(z)=∑k=0∞akzk,a0≠0be analytic in the unit disc. Any infinite complex vectorθ=(θ0,θ1,θ2,…)such that|θk|=1,k=0,1,2,…, induces a functionfθ(z)=∑k=0∞akθkzkwhich is still analytic in the unit disc.In this paper we study the problem of maximizing thep-means:∫02π|fθ(reiϕ)|pdϕover all possible vectorsθand for values ofrclose to0and for allp<2.It is proved that a maximizing function isf1(z)=−|a0|+∑k=1∞|ak|zkand thatrcould be taken to be any positive number which is smaller than the radius of the largest disc centered at the origin which can be inscribed in the zero sets off1. This problem is originated by a well known majorant problem for Fourier coefficients that was studied by Hardy and Littlewood.One consequence of our paper is that forp<2the extremal function for the Hardy-Littlewood problem should be−|a0|+∑k=1∞|ak|zk.We also give some applications to derive some sharp inequalities for the classes of Schlicht functions and of functions of positive real part.

scholarly journals The extremal function for partial bipartite tilings

2012 ◽  
Vol 33 (5) ◽  
pp. 807-815 ◽  
Author(s):  
Codruţ Grosu ◽  
Jan Hladký
Keyword(s):  
Extremal Function ◽  
The Extremal Function

Unpredictability of the coefficients of m-fold symmetric bi-starlike functions

2014 ◽  
Vol 25 (07) ◽  
pp. 1450064 ◽  
Author(s):  
Samaneh G. Hamidi ◽  
Jay M. Jahangiri
Keyword(s):  
Extremal Function ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Square Root ◽  
Koebe Function ◽  
The Extremal Function

In 1984, Libera and Zlotkiewicz proved that the inverse of the square-root transform of the Koebe function is the extremal function for the inverses of odd univalent functions. The purpose of this paper is to point out that this is not the case for the m-fold symmetric bi-starlike functions by demonstrating the unpredictability of the coefficients of such functions.

scholarly journals Convolutions of prestarlike functions

1983 ◽  
Vol 6 (1) ◽  
pp. 59-68 ◽  
Author(s):  
O. P. Ahuja ◽  
H. Silverman
Keyword(s):  
Unit Disk ◽  
Extremal Function ◽  
Extreme Points ◽  
Distortion Theorems ◽  
The Extremal Function ◽  
Class Of Functions

The convolution of two functionsf(z)=∑n=0∞anznandg(z)=∑n=0∞bnzndefined as(f∗g)(z)=∑n=0∞anbnzn. Forf(z)=z−∑n=2∞anznandg(z)=z/(1−z)2(1−γ), the extremal function for the class of functions starlike of orderγ, we investigate functionsh, whereh(z)=(f∗g)(z), which satisfy the inequality|(zh′/h)−1|/|(zh′/h)+(1-2α)|<β,0≤α<1,0<β≤1for allzin the unit disk. Such functionsfare said to beγ-prestarlike of orderαand typeβ. We characterize this family in terms of its coefficients, and then determine extreme points, distortion theorems, and radii of univalence, starlikeness, and convexity. All results are sharp.

scholarly journals Minor-monotone crossing number

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Drago Bokal ◽  
Gašper Fijavž ◽  
Bojan Mohar
Keyword(s):  
Topological Structure ◽  
Crossing Number ◽  
Graph Invariant ◽  
Basic Properties ◽  
International Audience ◽  
Graph Families ◽  
A Minor ◽  
Monotone Graph

International audience The minor crossing number of a graph $G$, $rmmcr(G)$, is defined as the minimum crossing number of all graphs that contain $G$ as a minor. We present some basic properties of this new minor-monotone graph invariant. We give estimates on mmcr for some important graph families using the topological structure of graphs satisfying \$mcr(G) ≤k$.

scholarly journals Note on Schiffer’s Variation in the Class of Univalent Functions in the Unit Disc

1968 ◽  
Vol 32 ◽  
pp. 273-276
Author(s):  
Kikuji Matsumoto
Keyword(s):  
Unit Disc ◽  
Extremal Function ◽  
Univalent Functions ◽  
The Real ◽  
Maximum Value ◽  
The Extremal Function

Let S denote the class of univalent functions f(z) in the unit disc D: | z | < 1 with the following expansion: (1) f(z) = z + a2z2 + a3z3 + · · · · anzn + · ··.We denote by fn(z) the extremal function in S which gives the maximum value of the real part of an and by Dn the image of D under w = fn(z).

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