The signless Laplacian spectral radius of<i>k</i>-connected irregular graphs

Shiu, Wai Chee; Huang, Peng; Sun, Pak Kiu

doi:10.1080/03081087.2016.1209731

Cited by 10 publications

(8 citation statements)

References 11 publications

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“…The authors in [20] also indicated that when k ≥ √ n, the bound in (1.9) is better than the bound in (1.8) and with the same arguments they improved the bound in (1.8), which were given in their remarks.…”

Section: Tr(mentioning

confidence: 67%

On the Largest Distance (Signless Laplacian) Eigenvalue of Non-transmission-regular Graphs

Liu

Shu

Xue

2018

ELA

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Let $G=(V(G),E(G))$ be a $k$-connected graph with $n$ vertices and $m$ edges. Let $D(G)$ be the distance matrix of $G$. Suppose $\lambda_1(D)\geq \cdots \geq \lambda_n(D)$ are the $D$-eigenvalues of $G$. The transmission of $v_i \in V(G)$, denoted by $Tr_G(v_i)$ is defined to be the sum of distances from $v_i$ to all other vertices of $G$, i.e., the row sum $D_{i}(G)$ of $D(G)$ indexed by vertex $v_i$ and suppose that $D_1(G)\geq \cdots \geq D_n(G)$. The $Wiener~ index$ of $G$ denoted by $W(G)$ is given by $W(G)=\frac{1}{2}\sum_{i=1}^{n}D_i(G)$. Let $Tr(G)$ be the $n\times n$ diagonal matrix with its $(i,i)$-entry equal to $TrG(v_i)$. The distance signless Laplacian matrix of $G$ is defined as $D^Q(G)=Tr(G)+D(G)$ and its spectral radius is denoted by $\rho_1(D^Q(G))$ or $\rho_1$. A connected graph $G$ is said to be $t$-transmission-regular if $Tr_G(v_i) =t$ for every vertex $v_i\in V(G)$, otherwise, non-transmission-regular. In this paper, we respectively estimate $D_1(G)-\lambda_1(G)$ and $2D_1(G)-\rho_1(G)$ for a $k$-connected non-transmission-regular graph in different ways and compare these obtained results. And we conjecture that $D_1(G)-\lambda_1(G)>\frac{1}{n+1}$. Moreover, we show that the conjecture is valid for trees.

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Section: Tr(mentioning

confidence: 67%

On the Largest Distance (Signless Laplacian) Eigenvalue of Non-transmission-regular Graphs

Liu

Shu

Xue

2018

ELA

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“…The argument in [6,30] leads easily to the following result. For completeness, however, we include a proof here.…”

Section: Bounds For the α-Spectral Radius Of Irregular Graphsmentioning

confidence: 89%

“…For a k-connected irregular graph G on n ≥ 3 vertices with m edges and maximum degree ∆, Chen and Hou [6] (see also Shiu et al [30]) showed that…”

Section: Bounds For the α-Spectral Radius Of Irregular Graphsmentioning

confidence: 99%

On the α-spectral radius of graphs

Guo¹,

Zhou

2020

Appl Anal Discrete Math

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For 0 ? ? ? 1, Nikiforov proposed to study the spectral properties of the family of matrices A?(G) = ?D(G)+(1 ? ?)A(G) of a graph G, where D(G) is the degree diagonal matrix and A(G) is the adjacency matrix of G. The ?-spectral radius of G is the largest eigenvalue of A?(G). For a graph with two pendant paths at a vertex or at two adjacent vertices, we prove results concerning the behavior of the ?-spectral radius under relocation of a pendant edge in a pendant path. We give upper bounds for the ?-spectral radius for unicyclic graphs G with maximum degree ? ? 2, connected irregular graphs with given maximum degree and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with the second largest ?-spectral radius among trees, and the unique tree with the largest ?-spectral radius among trees with given diameter. We also determine the unique graphs so that the difference between the maximum degree and the ?-spectral radius is maximum among trees, unicyclic graphs and non-bipartite graphs, respectively.

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“…Ning et al [15] gave a lower bound on 2∆ − q 1 (G) for a connected irregular graph G in terms of the diameter. Shui et al [19] gave a lower bound on 2∆ − q 1 (G) and 2∆−q 1 (H) for a k-connected irregular graph G and a proper spanning subgraph H of a ∆-regular k-connected graph, respectively. Li et al [10] obtianed the lower bounds on ∆ − λ 1 (G) for irregular connected k-graphs in terms of vertex degrees, the diameter, and the number of vertices and edges.…”

Section: Introductionmentioning

confidence: 99%

“…Chen et al [3] presented several upper bounds on λ 1 (G) and q 1 (G) for a k-graph G in terms of degree sequences. We are inspired by two articles of Shui et al [19] and Li et al [10]. In this paper, we give the bounds of (signless Laplacian) spectral radius of subgraphs of f (-edge)-connected d-regular (linear) k-graphs.…”

Section: Introductionmentioning

confidence: 99%

The (signless Laplacian) spectral radius (of subgraphs) of uniform hypergraphs

Duan

Wang

Xiao

et al. 2019

Filomat

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Let λ 1 (G) and q 1 (G) be the spectral radius and the signless Laplacian spectral radius of a kuniform hypergraph G, respectively. In this paper, we give the lower bounds of d − λ 1 (H) and 2d − q 1 (H), where H is a proper subgraph of a f (-edge)-connected d-regular (linear) k-uniform hypergraph. Meanwhile, we also give the lower bounds of 2∆ − q 1 (G) and ∆ − λ 1 (G), where G is a nonregular f (-edge)-connected (linear) k-uniform hypergraph with maximum degree ∆.

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The signless Laplacian spectral radius ofk-connected irregular graphs

Cited by 10 publications

References 11 publications

On the Largest Distance (Signless Laplacian) Eigenvalue of Non-transmission-regular Graphs

On the Largest Distance (Signless Laplacian) Eigenvalue of Non-transmission-regular Graphs

On the α-spectral radius of graphs

The (signless Laplacian) spectral radius (of subgraphs) of uniform hypergraphs

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