Recognition of characteristically simple group $A_5\times A_5$ by character degree graph and order

Maryam Khademi; Behrooz Khosravi

Displaying similar documents to “Recognition of characteristically simple group $A_{5} \times A_{5}$ by character degree graph and order”

Characterization by intersection graph of some families of finite nonsimple groups

Hossein Shahsavari, Behrooz Khosravi (2021)

Czechoslovak Mathematical Journal

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For a finite group $G$ , $Γ (G)$ , the intersection graph of $G$ , is a simple graph whose vertices are all nontrivial proper subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent when $H \cap K \neq 1$ . In this paper, we classify all finite nonsimple groups whose intersection graphs have a leaf and also we discuss the characterizability of them using their intersection graphs.

On the intersection graph of a finite group

Hossein Shahsavari, Behrooz Khosravi (2017)

Czechoslovak Mathematical Journal

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For a finite group $G$ , the intersection graph of $G$ which is denoted by $Γ (G)$ is an undirected graph such that its vertices are all nontrivial proper subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent when $H \cap K \neq 1$ . In this paper we classify all finite groups whose intersection graphs are regular. Also, we find some results on the intersection graphs of simple groups and finally we study the structure of $Aut (Γ (G))$ .

Degree sums of adjacent vertices for traceability of claw-free graphs

Tao Tian, Liming Xiong, Zhi-Hong Chen, Shipeng Wang (2022)

Czechoslovak Mathematical Journal

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The line graph of a graph $G$ , denoted by $L (G)$ , has $E (G)$ as its vertex set, where two vertices in $L (G)$ are adjacent if and only if the corresponding edges in $G$ have a vertex in common. For a graph $H$ , define ${\bar{σ}}_{2} (H) = min {d (u) + d (v) : u v \in E (H)}$ . Let $H$ be a 2-connected claw-free simple graph of order $n$ with $δ (H) \geq 3$ . We show that, if ${\bar{σ}}_{2} (H) \geq \frac{1}{7} (2 n - 5)$ and $n$ is sufficiently large, then either $H$ is traceable or the Ryjáček’s closure $cl (H) = L (G)$ , where $G$ is an essentially $2$ -edge-connected triangle-free graph that can be contracted to one of the two graphs of order 10...

On the diameter of the intersection graph of a finite simple group

Xuanlong Ma (2016)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group. The intersection graph $Δ_{G}$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of $G$ , and two distinct vertices $X$ and $Y$ are adjacent if $X \cap Y \neq 1$ , where $1$ denotes the trivial subgroup of order $1$ . A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters...

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 $e_{2}$ ...

Edge-colouring of graphs and hereditary graph properties

Samantha Dorfling, Tomáš Vetrík (2016)

Czechoslovak Mathematical Journal

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Edge-colourings of graphs have been studied for decades. We study edge-colourings with respect to hereditary graph properties. For a graph $G$ , a hereditary graph property $𝒫$ and $l \geq 1$ we define $χ_{𝒫, l}^{'} (G)$ to be the minimum number of colours needed to properly colour the edges of $G$ , such that any subgraph of $G$ induced by edges coloured by (at most) $l$ colours is in $𝒫$ . We present a necessary and sufficient condition for the existence of $χ_{𝒫, l}^{'} (G)$ . We focus on edge-colourings of graphs with respect to the hereditary...

On the multiplicity of Laplacian eigenvalues for unicyclic graphs

Fei Wen, Qiongxiang Huang (2022)

Czechoslovak Mathematical Journal

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Let $G$ be a connected graph of order $n$ and $U$ a unicyclic graph with the same order. We firstly give a sharp bound for $m_{G} (μ)$ , the multiplicity of a Laplacian eigenvalue $μ$ of $G$ . As a straightforward result, $m_{U} (1) \leq n - 2$ . We then provide two graph operations (i.e., grafting and shifting) on graph $G$ for which the value of $m_{G} (1)$ is nondecreasing. As applications, we get the distribution of $m_{U} (1)$ for unicyclic graphs on $n$ vertices. Moreover, for the two largest possible values of $m_{U} (1) \in {n - 5, n - 3}$ , the corresponding graphs $U$ are...

Saturation numbers for linear forests $P_{6} + t P_{2}$

Jingru Yan (2023)

Czechoslovak Mathematical Journal

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A graph $G$ is $H$ -saturated if it contains no $H$ as a subgraph, but does contain $H$ after the addition of any edge in the complement of $G$ . The saturation number, $sat (n, H)$ , is the minimum number of edges of a graph in the set of all $H$ -saturated graphs of order $n$ . We determine the saturation number $sat (n, P_{6} + t P_{2})$ for $n \geq \frac{10}{3} t + 10$ and characterize the extremal graphs for $n > \frac{10}{3} t + 20$ .

Some results on the co-intersection graph of submodules of a module

Lotf Ali Mahdavi, Yahya Talebi (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let $R$ be a ring with identity and $M$ be a unitary left $R$ -module. The co-intersection graph of proper submodules of $M$ , denoted by $Ω (M)$ , is an undirected simple graph whose vertex set $V (Ω)$ is a set of all nontrivial submodules of $M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $N + K \neq M$ . We study the connectivity, the core and the clique number of $Ω (M)$ . Also, we provide some conditions on the module $M$ , under which the clique number of $Ω (M)$ is infinite and $Ω (M)$ is a planar graph. Moreover, we give...

Note on improper coloring of $1$ -planar graphs

Yanan Chu, Lei Sun, Jun Yue (2019)

Czechoslovak Mathematical Journal

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A graph $G = (V, E)$ is called improperly $(d_{1}, \dots, d_{k})$ -colorable if the vertex set $V$ can be partitioned into subsets $V_{1}, \dots, V_{k}$ such that the graph $G [V_{i}]$ induced by the vertices of $V_{i}$ has maximum degree at most $d_{i}$ for all $1 \leq i \leq k$ . In this paper, we mainly study the improper coloring of $1$ -planar graphs and show that $1$ -planar graphs with girth at least $7$ are $(2, 0, 0, 0)$ -colorable.

Groups satisfying the two-prime hypothesis with a composition factor isomorphic to PSL $_{2} (q)$ for $q \geq 7$

Mark L. Lewis, Yanjun Liu, Hung P. Tong-Viet (2018)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group and write $cd (G)$ for the degree set of the complex irreducible characters of $G$ . The group $G$ is said to satisfy the two-prime hypothesis if for any distinct degrees $a, b \in cd (G)$ , the total number of (not necessarily different) primes of the greatest common divisor $gcd (a, b)$ is at most $2$ . We prove an upper bound on the number of irreducible character degrees of a nonsolvable group that has a composition factor isomorphic to PSL $_{2} (q)$ for $q \geq 7$ .

On sets of discontinuities of functions continuous on all lines

Luděk Zajíček (2022)

Commentationes Mathematicae Universitatis Carolinae

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Answering a question asked by K. C. Ciesielski and T. Glatzer in 2013, we construct a $C^{1}$ -smooth function $f$ on $[0, 1]$ and a closed set $M \subset graph f$ nowhere dense in $graph f$ such that there does not exist any linearly continuous function on $ℝ^{2}$ (i.e., function continuous on all lines) which is discontinuous at each point of $M$ . We substantially use a recent full characterization of sets of discontinuity points of linearly continuous functions on $ℝ^{n}$ proved by T. Banakh and O. Maslyuchenko in 2020. As an easy consequence...

Turán number of two vertex-disjoint copies of cliques

Caiyun Hu (2024)

Czechoslovak Mathematical Journal

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The Turán number of a given graph $H$ , denoted by $ex (n, H)$ , is the maximum number of edges in an $H$ -free graph on $n$ vertices. Applying a well-known result of Hajnal and Szemerédi, we determine the Turán number $ex (n, K_{p} \cup K_{q}$ ) of a vertex-disjoint union of cliques $K_{p}$ and $K_{q}$ for all values of $n$ .

The extremal irregularity of connected graphs with given number of pendant vertices

Xiaoqian Liu, Xiaodan Chen, Junli Hu, Qiuyun Zhu (2022)

Czechoslovak Mathematical Journal

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The irregularity of a graph $G = (V, E)$ is defined as the sum of imbalances $| d_{u} - d_{v} |$ over all edges $u v \in E$ , where $d_{u}$ denotes the degree of the vertex $u$ in $G$ . This graph invariant, introduced by Albertson in 1997, is a measure of the defect of regularity of a graph. In this paper, we completely determine the extremal values of the irregularity of connected graphs with $n$ vertices and $p$ pendant vertices ( $1 \leq p \leq n - 1$ ), and characterize the corresponding extremal graphs.

Displaying similar documents to “Recognition of characteristically simple group A 5 × A 5 by character degree graph and order”

Displaying similar documents to “Recognition of characteristically simple group $A_{5} \times A_{5}$ by character degree graph and order”