Displaying similar documents to “The small Ree group 2 G 2 ( 3 2 n + 1 ) and related graph”

The Turán number of the graph 3 P 4

Halina Bielak, Sebastian Kieliszek (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let e x ( n , G ) denote the maximum number of edges in a graph on n vertices which does not contain G as a subgraph. Let P i denote a path consisting of i vertices and let m P i denote m disjoint copies of P i . In this paper we count e x ( n , 3 P 4 ) .

Recognizability of finite groups by Suzuki group

Alireza Khalili Asboei, Seyed Sadegh Salehi Amiri (2019)

Archivum Mathematicum

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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 S z ( q ) if and only if 𝒮 ( G ) 𝒮 ( S z ( q ) ) , where q = 2 2 m + 1 8 .

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...

Characterizing finite groups whose enhanced power graphs have universal vertices

David G. Costanzo, Mark L. Lewis, Stefano Schmidt, Eyob Tsegaye, Gabe Udell (2024)

Czechoslovak Mathematical Journal

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Let G be a finite group and construct a graph Δ ( G ) by taking G { 1 } as the vertex set of Δ ( G ) and by drawing an edge between two vertices x and y if x , y is cyclic. Let K ( G ) be the set consisting of the universal vertices of Δ ( G ) along the identity element. For a solvable group G , we present a necessary and sufficient condition for K ( G ) to be nontrivial. We also develop a connection between Δ ( G ) and K ( G ) when | G | is divisible by two distinct primes and the diameter of Δ ( G ) is 2.

Even factor of bridgeless graphs containing two specified edges

Nastaran Haghparast, Dariush Kiani (2018)

Czechoslovak Mathematical Journal

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An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. Let G be a bridgeless simple graph with minimum degree at least 3 . Jackson and Yoshimoto (2007) showed that G has an even factor containing two arbitrary prescribed edges. They also proved that G has an even factor in which each component has order at least four. Moreover, Xiong, Lu and Han (2009) showed that for each pair of edges e 1 and e 2 of G , there is an even factor containing e 1 and e 2 ...

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...

On upper bounds for total k -domination number via the probabilistic method

Saylí Sigarreta, Saylé Sigarreta, Hugo Cruz-Suárez (2023)

Kybernetika

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For a fixed positive integer k and G = ( V , E ) a connected graph of order n , whose minimum vertex degree is at least k , a set S V is a total k -dominating set, also known as a k -tuple total dominating set, if every vertex v V has at least k neighbors in S . The minimum size of a total k -dominating set for G is called the total k -domination number of G , denoted by γ k t ( G ) . The total k -domination problem is to determine a minimum total k -dominating set of G . Since the exact problem is in general quite difficult...

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 ) ) ....

A note on the double Roman domination number of graphs

Xue-Gang Chen (2020)

Czechoslovak Mathematical Journal

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For a graph G = ( V , E ) , a double Roman dominating function is a function f : V { 0 , 1 , 2 , 3 } having the property that if f ( v ) = 0 , then the vertex v must have at least two neighbors assigned 2 under f or one neighbor with f ( w ) = 3 , and if f ( v ) = 1 , then the vertex v must have at least one neighbor with f ( w ) 2 . The weight of a double Roman dominating function f is the sum f ( V ) = v V f ( v ) . The minimum weight of a double Roman dominating function on G is called the double Roman domination number of G and is denoted by γ dR ( G ) . In this paper, we establish a new...

Distance matrices perturbed by Laplacians

Balaji Ramamurthy, Ravindra Bhalchandra Bapat, Shivani Goel (2020)

Applications of Mathematics

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Let T be a tree with n vertices. To each edge of T we assign a weight which is a positive definite matrix of some fixed order, say, s . Let D i j denote the sum of all the weights lying in the path connecting the vertices i and j of T . We now say that D i j is the distance between i and j . Define D : = [ D i j ] , where D i i is the s × s null matrix and for i j , D i j is the distance between i and j . Let G be an arbitrary connected weighted graph with n vertices, where each weight is a positive definite matrix of order...

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...