scholarly journals Identifying Graph Automorphisms Using Determining Sets

10.37236/1104 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Debra L. Boutin
Keyword(s):  
Lower Bounds ◽  
Sharp Bounds ◽  
Determining Set ◽  
Kneser Graphs

A set of vertices $S$ is a determining set for a graph $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The determining number of a graph is the size of a smallest determining set. This paper describes ways of finding and verifying determining sets, gives natural lower bounds on the determining number, and shows how to use orbits to investigate determining sets. Further, determining sets of Kneser graphs are extensively studied, sharp bounds for their determining numbers are provided, and all Kneser graphs with determining number $2$, $3,$ or $4$ are given.

Sharp Localized Inequalities for Fourier Multipliers

2014 ◽  
Vol 66 (6) ◽  
pp. 1358-1381
Author(s):  
Adam Osękowski
Keyword(s):  
Lower Bounds ◽  
Lévy Processes ◽  
Levy Processes ◽  
Fourier Multipliers ◽  
Probabilistic Methods ◽  
Independent Interest ◽  
Probability Measures ◽  
Sharp Bounds ◽  
The Real

AbstractIn this paper we study sharp localized Lq → Lp estimates for Fourier multipliers resulting from modulation of the jumps of Lévy processes. The proofs of these estimates rest on probabilistic methods and exploit related sharp bounds for differentially subordinated martingales, which are of independent interest. The lower bounds for the constants involve the analysis of laminates, a family of certain special probability measures on 2×2 matrices. As an application, we obtain new sharp bounds for the real and imaginary parts of the Beurling–Ahlfors operator.

scholarly journals Sharp Bounds of the Hyper-Zagreb Index on Acyclic, Unicylic, and Bicyclic Graphs

2017 ◽  
Vol 2017 ◽  
pp. 1-5 ◽  
Author(s):  
Wei Gao ◽  
Muhammad Kamran Jamil ◽  
Aisha Javed ◽  
Mohammad Reza Farahani ◽  
Shaohui Wang ◽  
...  
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Sharp Bounds ◽  
Zagreb Indices ◽  
Graph Transformations ◽  
Mathematical Properties ◽  
Bicyclic Graphs ◽  
Important Branch

The hyper-Zagreb index is an important branch in the Zagreb indices family, which is defined as∑uv∈E(G)‍(d(u)+d(v))2, whered(v)is the degree of the vertexvin a graphG=(V(G),E(G)). In this paper, the monotonicity of the hyper-Zagreb index under some graph transformations was studied. Using these nice mathematical properties, the extremal graphs amongn-vertex trees (acyclic), unicyclic, and bicyclic graphs are determined for hyper-Zagreb index. Furthermore, the sharp upper and lower bounds on the hyper-Zagreb index of these graphs are provided.

scholarly journals Sharp Bounds of the Hermitian Toeplitz Determinants for Some Classes of Close-to-Convex Functions

2021 ◽  
Author(s):  
Adam Lecko ◽  
Barbara Śmiarowska
Keyword(s):  
Lower Bounds ◽  
Convex Functions ◽  
Upper And Lower Bounds ◽  
Toeplitz Determinants ◽  
Sharp Bounds

AbstractSharp upper and lower bounds of the Hermitian Toeplitz determinants of the second and third orders are found for various subclasses of close-to-convex functions.

scholarly journals Sharp bounds for a ratio of the q-gamma function in terms of the q-digamma function

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Faris Alzahrani ◽  
Ahmed Salem ◽  
Moustafa El-Shahed
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Gamma Function ◽  
Open Problem ◽  
Upper And Lower Bounds ◽  
Digamma Function ◽  
Sharp Bounds ◽  
Gamma Functions ◽  
Numer Math

AbstractIn the present paper, we introduce sharp upper and lower bounds to the ratio of two q-gamma functions ${\Gamma }_{q}(x+1)/{\Gamma }_{q}(x+s)$ Γ q ( x + 1 ) / Γ q ( x + s ) for all real number s and $0< q\neq1$ 0 < q ≠ 1 in terms of the q-digamma function. Our results refine the results of Ismail and Muldoon (Internat. Ser. Numer. Math., vol. 119, pp. 309–323, 1994) and give the answer to the open problem posed by Alzer (Math. Nachr. 222(1):5–14, 2001). Also, for the classical gamma function, our results give a Kershaw inequality for all $0< s<1$ 0 < s < 1 when letting $q\to 1$ q → 1 and a new inequality for all $s>1$ s > 1 .

scholarly journals Sharp bounds for partition dimension of generalized Möbius ladders

2018 ◽  
Vol 16 (1) ◽  
pp. 1283-1290 ◽  
Author(s):  
Zafar Hussain ◽  
Junaid Alam Khan ◽  
Mobeen Munir ◽  
Muhammad Shoaib Saleem ◽  
Zaffar Iqbal
Keyword(s):  
Lower Bounds ◽  
Metric Space ◽  
Robot Navigation ◽  
Upper Bounds ◽  
Pivotal Role ◽  
Resolving Set ◽  
Sharp Bounds ◽  
Partition Dimension

AbstractThe concept of minimal resolving partition and resolving set plays a pivotal role in diverse areas such as robot navigation, networking, optimization, mastermind games and coin weighing. It is hard to compute exact values of partition dimension for a graphic metric space, (G, dG) and networks. In this article, we give the sharp upper bounds and lower bounds for the partition dimension of generalized Möbius ladders, Mm, n, for all n≥3 and m≥2.

ON DYNAMIC MONOPOLIES OF GRAPHS WITH PROBABILISTIC THRESHOLDS

2014 ◽  
Vol 90 (3) ◽  
pp. 363-375 ◽  
Author(s):  
HOSSEIN SOLTANI ◽  
MANOUCHEHR ZAKER
Keyword(s):  
Social Networks ◽  
Lower Bounds ◽  
Random Variable ◽  
Mean Value ◽  
Sharp Bounds ◽  
Dynamic Monopoly ◽  
Underlying Network ◽  
Spread Of Influence ◽  
Probabilistic Thresholds

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a graph and ${{\tau }}$ be an assignment of nonnegative thresholds to the vertices of $G$. A subset of vertices, $D$, is an irreversible dynamic monopoly of $(G, \tau )$ if the vertices of $G$ can be partitioned into subsets $D_0, D_1, \ldots, D_k$ such that $D_0=D$ and, for all $i$ with $0 \leq i \leq k-1$, each vertex $v$ in $D_{i+1}$ has at least $\tau (v)$ neighbours in the union of $D_0, D_1, \ldots, D_i$. Dynamic monopolies model the spread of influence or propagation of opinion in social networks, where the graph $G$ represents the underlying network. The smallest cardinality of any dynamic monopoly of $(G,\tau )$ is denoted by $\mathrm{dyn}_{\tau }(G)$. In this paper we assume that the threshold of each vertex $v$ of the network is a random variable $X_v$ such that $0\leq X_v \leq \deg _G(v)+1$. We obtain sharp bounds on the expectation and the concentration of $\mathrm{dyn}_{\tau }(G)$ around its mean value. We also obtain some lower bounds for the size of dynamic monopolies in terms of the order of graph and expectation of the thresholds.

scholarly journals Sharp bounds on the signless Laplacian Estrada index of graphs

Filomat ◽  
2014 ◽  
Vol 28 (10) ◽  
pp. 1983-1988 ◽  
Author(s):  
Shan Gao ◽  
Huiqing Liu
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Laplacian Matrix ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Sharp Bounds ◽  
Estrada Index ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian ◽  
The First Zagreb Index

Let G be a connected graph with n vertices and m edges. Let q1, q2,..., qn be the eigenvalues of the signless Laplacian matrix of G, where q1 ? q2 ? ... ? qn. The signless Laplacian Estrada index of G is defined as SLEE(G) = nPi=1 eqi. In this paper, we present some sharp lower bounds for SLEE(G) in terms of the k-degree and the first Zagreb index, respectively.

scholarly journals The determining number of Kneser graphs

2013 ◽  
Vol Vol. 15 no. 1 (Graph Theory) ◽  
Author(s):  
José Cáceres ◽  
Delia Garijo ◽  
Antonio González ◽  
Alberto Márquez ◽  
Marıa Luz Puertas
Keyword(s):  
Graph Theory ◽  
Group Theory ◽  
Symmetric Group ◽  
Minimum Cardinality ◽  
Wide Range ◽  
Base Size ◽  
International Audience ◽  
Determining Set ◽  
Kneser Graphs

Graph Theory International audience A set of vertices S is a determining set of a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of G is the minimum cardinality of a determining set of G. This paper studies the determining number of Kneser graphs. First, we compute the determining number of a wide range of Kneser graphs, concretely Kn:k with n≥k(k+1) / 2+1. In the language of group theory, these computations provide exact values for the base size of the symmetric group Sn acting on the k-subsets of 1,..., n. Then, we establish for which Kneser graphs Kn:k the determining number is equal to n-k, answering a question posed by Boutin. Finally, we find all Kneser graphs with fixed determining number 5, extending the study developed by Boutin for determining number 2, 3 or 4.

scholarly journals Sharp Bounds on the Minimum M-Eigenvalue of Elasticity M-Tensors

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 250
Author(s):  
Ying Zhang ◽  
Linxuan Sun ◽  
Gang Wang
Keyword(s):  
Lower Bounds ◽  
Elastic Material ◽  
Upper Bound ◽  
Nonlinear Elastic Material ◽  
Sharp Bounds ◽  
Numerical Examples ◽  
Material Analysis ◽  
Nonlinear Elastic

The M-eigenvalue of elasticity M-tensors play important roles in nonlinear elastic material analysis. In this paper, we establish an upper bound and two sharp lower bounds for the minimum M-eigenvalue of elasticity M-tensors without irreducible conditions, which improve some existing results. Numerical examples are proposed to verify the efficiency of the obtained results.

scholarly journals Using Determining Sets to Distinguish Kneser Graphs

10.37236/938 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Michael O. Albertson ◽  
Debra L. Boutin
Keyword(s):  
Kneser Graph ◽  
Nontrivial Automorphism ◽  
Vertex Set ◽  
Determining Set ◽  
Kneser Graphs

This work introduces the technique of using a carefully chosen determining set to prove the existence of a distinguishing labeling using few labels. A graph $G$ is said to be $d$-distinguishable if there is a labeling of the vertex set using $1, \ldots, d$ so that no nontrivial automorphism of $G$ preserves the labels. A set of vertices $S\subseteq V(G)$ is a determining set for $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. We prove that a graph is $d$-distinguishable if and only if it has a determining set that can be $(d-1)$-distinguished. We use this to prove that every Kneser graph $K_{n:k}$ with $n\geq 6$ and $k\geq 2$ is $2$-distinguishable.

Sign in / Sign up

Export Citation Format

Share Document