On distance Laplacian energy in terms of graph invariants

Hilal A. Ganie; Rezwan Ul Shaban; Bilal A. Rather; Shariefuddin Pirzada

Displaying similar documents to “On distance Laplacian energy in terms of graph invariants”

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...

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...

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...

Edit distance measure for graphs

Tomasz Dzido, Krzysztof Krzywdziński (2015)

Czechoslovak Mathematical Journal

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In this paper, we investigate a measure of similarity of graphs similar to the Ramsey number. We present values and bounds for $g (n, l)$ , the biggest number $k$ guaranteeing that there exist $l$ graphs on $n$ vertices, each two having edit distance at least $k$ . By edit distance of two graphs $G$ , $F$ we mean the number of edges needed to be added to or deleted from graph $G$ to obtain graph $F$ . This new extremal number $g (n, l)$ is closely linked to the edit distance of graphs. Using probabilistic methods we show...

Note on a conjecture for the sum of signless Laplacian eigenvalues

Xiaodan Chen, Guoliang Hao, Dequan Jin, Jingjian Li (2018)

Czechoslovak Mathematical Journal

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For a simple graph $G$ on $n$ vertices and an integer $k$ with $1 \leq k \leq n$ , denote by $𝒮_{k}^{+} (G)$ the sum of $k$ largest signless Laplacian eigenvalues of $G$ . It was conjectured that $𝒮_{k}^{+} (G) \leq e (G) + (\binom{k + 1}{2})$ , where $e (G)$ is the number of edges of $G$ . This conjecture has been proved to be true for all graphs when $k \in {1, 2, n - 1, n}$ , and for trees, unicyclic graphs, bicyclic graphs and regular graphs (for all $k$ ). In this note, this conjecture is proved to be true for all graphs when $k = n - 2$ , and for some new classes of graphs.

A spectral bound for graph irregularity

Felix Goldberg (2015)

Czechoslovak Mathematical Journal

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The imbalance of an edge $e = {u, v}$ in a graph is defined as $i (e) = | d (u) - d (v) |$ , where $d (\cdot)$ is the vertex degree. The irregularity $I (G)$ of $G$ is then defined as the sum of imbalances over all edges of $G$ . This concept was introduced by Albertson who proved that $I (G) \leq 4 n^{3} / 27$ (where $n = | V (G) |$ ) and obtained stronger bounds for bipartite and triangle-free graphs. Since then a number of additional bounds were given by various authors. In this paper we prove a new upper bound, which improves a bound found by Zhou and Luo in 2008. Our bound involves...

Partial sum of eigenvalues of random graphs

Israel Rocha (2020)

Applications of Mathematics

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Let $G$ be a graph on $n$ vertices and let $λ_{1} \geq λ_{2} \geq ... \geq λ_{n}$ be the eigenvalues of its adjacency matrix. For random graphs we investigate the sum of eigenvalues $s_{k} = \sum_{i = 1}^{k} λ_{i}$ , for $1 \leq k \leq n$ , and show that a typical graph has $s_{k} \leq (e (G) + k^{2}) / {(0.99 n)}^{1 / 2}$ , where $e (G)$ is the number of edges of $G$ . We also show bounds for the sum of eigenvalues within a given range in terms of the number of edges. The approach for the proofs was first used in Rocha (2020) to bound the partial sum of eigenvalues of the Laplacian matrix.

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 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 spectral determinations of the connected multicone graphs $K_{w} ▽ m P_{17}$ and $K_{w} ▽ m S$

Ali Zeydi Abdian, S. Morteza Mirafzal (2018)

Czechoslovak Mathematical Journal

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Finding and discovering any class of graphs which are determined by their spectra is always an important and interesting problem in the spectral graph theory. The main aim of this study is to characterize two classes of multicone graphs which are determined by both their adjacency and Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let $K_{w}$ denote a complete graph on $w$ vertices, and let $m$ be a positive integer number. In A. Z. Abdian (2016)...