EUDML | The spectral determinations of the connected multicone graphs $ K_w\bigtriangledown mP_{17} $ and $ K_w\bigtriangledown mS $

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Displaying similar documents to “The spectral determinations of the connected multicone graphs $K_{w} ▽ m P_{17}$ and $K_{w} ▽ m S$ ”

On 𝓕-independence in graphs

Frank Göring, Jochen Harant, Dieter Rautenbach, Ingo Schiermeyer (2009)

Discussiones Mathematicae Graph Theory

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Let be a set of graphs and for a graph G let $α_{} (G)$ and $α *_{} (G)$ denote the maximum order of an induced subgraph of G which does not contain a graph in as a subgraph and which does not contain a graph in as an induced subgraph, respectively. Lower bounds on $α_{} (G)$ and $α *_{} (G)$ are presented.

The fan graph is determined by its signless Laplacian spectrum

Muhuo Liu, Yuan Yuan, Kinkar Chandra Das (2020)

Czechoslovak Mathematical Journal

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Given a graph $G$ , if there is no nonisomorphic graph $H$ such that $G$ and $H$ have the same signless Laplacian spectra, then we say that $G$ is $Q$ -DS. In this paper we show that every fan graph $F_{n}$ is $Q$ -DS, where $F_{n} = K_{1} \lor P_{n - 1}$ and $n \geq 3$ .

Remarks on $D$ -integral complete multipartite graphs

Pavel Híc, Milan Pokorný (2016)

Czechoslovak Mathematical Journal

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A graph is called distance integral (or $D$ -integral) if all eigenvalues of its distance matrix are integers. In their study of $D$ -integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on $D$ -integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs $K_{p_{1}, p_{2}, p_{3}}$ with $p_{1} < p_{2} < p_{3}$ , and $K_{p_{1}, p_{2}, p_{3}, p_{4}}$ with $p_{1} < p_{2} < p_{3} < p_{4}$ , as well as the infinite classes of distance integral...

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...

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 $Aut (Γ (G))$ .

Generalized connectivity of some total graphs

Yinkui Li, Yaping Mao, Zhao Wang, Zongtian Wei (2021)

Czechoslovak Mathematical Journal

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We study the generalized $k$ -connectivity $κ_{k} (G)$ as introduced by Hager in 1985, as well as the more recently introduced generalized $k$ -edge-connectivity $λ_{k} (G)$ . We determine the exact value of $κ_{k} (G)$ and $λ_{k} (G)$ for the line graphs and total graphs of trees, unicyclic graphs, and also for complete graphs for the case $k = 3$ .

On distance Laplacian energy in terms of graph invariants

Hilal A. Ganie, Rezwan Ul Shaban, Bilal A. Rather, Shariefuddin Pirzada (2023)

Czechoslovak Mathematical Journal

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For a simple connected graph $G$ of order $n$ having distance Laplacian eigenvalues $ρ_{1}^{L} \geq ρ_{2}^{L} \geq \dots \geq ρ_{n}^{L}$ , the distance Laplacian energy $DLE (G)$ is defined as $DLE (G) = \sum_{i = 1}^{n} | ρ_{i}^{L} - 2 W (G) / n |$ , where $W (G)$ is the Wiener index of $G$ . We obtain a relationship between the Laplacian energy and the distance Laplacian energy for graphs with diameter 2. We obtain lower bounds for the distance Laplacian energy $DLE (G)$ in terms of the order $n$ , the Wiener index $W (G)$ , the independence number, the vertex connectivity number and other given parameters. We characterize the...

Graphs with small diameter determined by their $D$ -spectra

Ruifang Liu, Jie Xue (2018)

Czechoslovak Mathematical Journal

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Let $G$ be a connected graph with vertex set $V (G) = {v_{1}, v_{2}, ..., v_{n}}$ . The distance matrix $D (G) = {(d_{i j})}_{n \times n}$ is the matrix indexed by the vertices of $G$ , where $d_{i j}$ denotes the distance between the vertices $v_{i}$ and $v_{j}$ . Suppose that $λ_{1} (D) \geq λ_{2} (D) \geq \dots \geq λ_{n} (D)$ are the distance spectrum of $G$ . The graph $G$ is said to be determined by its $D$ -spectrum if with respect to the distance matrix $D (G)$ , any graph having the same spectrum as $G$ is isomorphic to $G$ . We give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs...

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.

Proper connection number of bipartite graphs

Jun Yue, Meiqin Wei, Yan Zhao (2018)

Czechoslovak Mathematical Journal

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An edge-colored graph $G$ is proper connected if every pair of vertices is connected by a proper path. The proper connection number of a connected graph $G$ , denoted by $pc (G)$ , is the smallest number of colors that are needed to color the edges of $G$ in order to make it proper connected. In this paper, we obtain the sharp upper bound for $pc (G)$ of a general bipartite graph $G$ and a series of extremal graphs. Additionally, we give a proper $2$ -coloring for a connected bipartite graph $G$ having $δ (G) \geq 2$ and a dominating...

On the signless Laplacian and normalized signless Laplacian spreads of graphs

Emina Milovanović, Serife B. Bozkurt Altindağ, Marjan Matejić, Igor Milovanović (2023)

Czechoslovak Mathematical Journal

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Let $G = (V, E)$ , $V = {v_{1}, v_{2}, ..., v_{n}}$ , be a simple connected graph with $n$ vertices, $m$ edges and a sequence of vertex degrees $d_{1} \geq d_{2} \geq \dots \geq d_{n}$ . Denote by $A$ and $D$ the adjacency matrix and diagonal vertex degree matrix of $G$ , respectively. The signless Laplacian of $G$ is defined as $L^{+} = D + A$ and the normalized signless Laplacian matrix as $ℒ^{+} = D^{- 1 / 2} L^{+} D^{- 1 / 2}$ . The normalized signless Laplacian spreads of a connected nonbipartite graph $G$ are defined as $r (G) = γ_{2}^{+} / γ_{n}^{+}$ and $l (G) = γ_{2}^{+} - γ_{n}^{+}$ , where $γ_{1}^{+} \geq γ_{2}^{+} \geq \dots \geq γ_{n}^{+} \geq 0$ are eigenvalues of $ℒ^{+}$ . We establish sharp lower and upper bounds for the normalized signless...

Unbalanced unicyclic and bicyclic graphs with extremal spectral radius

Francesco Belardo, Maurizio Brunetti, Adriana Ciampella (2021)

Czechoslovak Mathematical Journal

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A signed graph $Γ$ is a graph whose edges are labeled by signs. If $Γ$ has $n$ vertices, its spectral radius is the number $ρ (Γ) : = max {| λ_{i} (Γ) | : 1 \leq i \leq n}$ , where $λ_{1} (Γ) \geq \dots \geq λ_{n} (Γ)$ are the eigenvalues of the signed adjacency matrix $A (Γ)$ . Here we determine the signed graphs achieving the minimal or the maximal spectral radius in the classes $𝔘_{n}$ and $𝔅_{n}$ of unbalanced unicyclic graphs and unbalanced bicyclic graphs, respectively.