Displaying similar documents to “Inequalities for real number sequences with applications in spectral graph theory”

Some properties of generalized distance eigenvalues of graphs

Yuzheng Ma, Yan Ling Shao (2024)

Czechoslovak Mathematical Journal

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Let G be a simple connected graph with vertex set V ( G ) = { v 1 , v 2 , , v n } and edge set E ( G ) , and let d v i be the degree of the vertex v i . Let D ( G ) be the distance matrix and let T r ( G ) be the diagonal matrix of the vertex transmissions of G . The generalized distance matrix of G is defined as D α ( G ) = α T r ( G ) + ( 1 - α ) D ( G ) , where 0 α 1 . Let λ 1 ( D α ( G ) ) λ 2 ( D α ( G ) ) ... λ n ( D α ( G ) ) be the generalized distance eigenvalues of G , and let k be an integer with 1 k n . We denote by S k ( D α ( G ) ) = λ 1 ( D α ( G ) ) + λ 2 ( D α ( G ) ) + ... + λ k ( D α ( G ) ) the sum of the k largest generalized distance eigenvalues. The generalized distance spread of a graph G is defined as D α S ( G ) = λ 1 ( D α ( G ) ) - λ n ( D α ( G ) ) ....

Sum-product theorems and incidence geometry

Mei-Chu Chang, Jozsef Solymosi (2007)

Journal of the European Mathematical Society

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In this paper we prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P 1 , , P 4 , and Q 1 , , Q n 2 , if there are n ( 1 + δ ) / 2 many distinct lines between P i and Q j for all i , j , then P 1 , , P 4 are collinear. If the number of the distinct lines is < c n 1 / 2 then the cross ratio of the four points is algebraic. 2. Given c > 0 , there is δ > 0 such that for any P 1 , P 2 , P 3 2 noncollinear, and Q 1 , , Q n 2 , if there are c n 1 / 2 many distinct lines between P i and Q j for all i , j , then for any P 2 { P 1 , P 2 , P 3 } , we have δ n distinct lines between P and Q j . 3. Given...

On path-quasar Ramsey numbers

Binlong Li, Bo Ning (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let G 1 and G 2 be two given graphs. The Ramsey number R ( G 1 , G 2 ) is the least integer r such that for every graph G on r vertices, either G contains a G 1 or G ¯ contains a G 2 . Parsons gave a recursive formula to determine the values of R ( P n , K 1 , m ) , where P n is a path on n vertices and K 1 , m is a star on m + 1 vertices. In this note, we study the Ramsey numbers R ( P n , K 1 F m ) , where F m is a linear forest on m vertices. We determine the exact values of R ( P n , K 1 F m ) for the cases m n and m 2 n , and for the case that F m has no odd component. Moreover, we...

Involutivity degree of a distribution at superdensity points of its tangencies

Silvano Delladio (2021)

Archivum Mathematicum

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Let Φ 1 , ... , Φ k + 1 (with k 1 ) be vector fields of class C k in an open set U N + m , let 𝕄 be a N -dimensional C k submanifold of U and define 𝕋 : = { z 𝕄 : Φ 1 ( z ) , ... , Φ k + 1 ( z ) T z 𝕄 } where T z 𝕄 is the tangent space to 𝕄 at z . Then we expect the following property, which is obvious in the special case when z 0 is an interior point (relative to 𝕄 ) of 𝕋 : If z 0 𝕄 is a ( N + k ) -density point (relative to 𝕄 ) of 𝕋 then all the iterated Lie brackets of order less or equal to k Φ i 1 ( z 0 ) , [ Φ i 1 , Φ i 2 ] ( z 0 ) , [ [ Φ i 1 , Φ i 2 ] , Φ i 3 ] ( z 0 ) , ... ( h , i h k + 1 ) belong to T z 0 𝕄 . Such a property has been proved in [9] for k = 1 and its proof in the...

The potential-Ramsey number of K n and K t - k

Jin-Zhi Du, Jian Hua Yin (2022)

Czechoslovak Mathematical Journal

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A nonincreasing sequence π = ( d 1 , ... , d n ) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of π . Given two graphs G 1 and G 2 , A. Busch et al. (2014) introduced the potential-Ramsey number of G 1 and G 2 , denoted by r pot ( G 1 , G 2 ) , as the smallest nonnegative integer m such that for every m -term graphic sequence π , there is a realization G of π with G 1 G or with G 2 G ¯ , where G ¯ is the complement of G . For t 2 and 0 k t 2 , let K t - k be the graph...

The small Ree group 2 G 2 ( 3 2 n + 1 ) and related graph

Alireza K. Asboei, Seyed S. S. Amiri (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let G be a finite group. The main supergraph 𝒮 ( G ) is a graph with vertex set G in which two vertices x and y are adjacent if and only if o ( x ) o ( y ) or o ( y ) o ( x ) . In this paper, we will show that G 2 G 2 ( 3 2 n + 1 ) if and only if 𝒮 ( G ) 𝒮 ( 2 G 2 ( 3 2 n + 1 ) ) . As a main consequence of our result we conclude that Thompson’s problem is true for the small Ree group 2 G 2 ( 3 2 n + 1 ) .

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

Complex series and connected sets

B. Jasek

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CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO Σ 1 , Σ 2 , Σ 3 , Σ 4 INESSENTIAL RESTRICTIONOF GENERALITY ...............................................................................................................................................................