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

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4733-4745 ◽  
Author(s):  
Cunxiang Duan ◽  
Ligong Wang ◽  
Peng Xiao ◽  
Xihe Li
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Uniform Hypergraph ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Uniform Hypergraphs ◽  
Signless Laplacian ◽  
Proper Subgraph

Let ?1(G) and q1(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-?1(H) and 2d-q1(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?-q1(G) and ?-?1(G), where G is a nonregular f (-edge)-connected (linear) k-uniform hypergraph with maximum degree ?.

On the distance and distance signless Laplacian eigenvalues of graphs and the smallest Gersgorin disc

2018 ◽  
Vol 34 ◽  
pp. 191-204 ◽  
Author(s):  
Fouzul Atik ◽  
Pratima Panigrahi
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Nonnegative Matrix ◽  
Laplacian Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Spectral Radius ◽  
Laplacian Eigenvalues ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

The \emph{distance matrix} of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the $i$th and $j$th vertices of $G$. The \emph{distance signless Laplacian matrix} of the graph $G$ is $D_Q(G)=D(G)+Tr(G)$, where $Tr(G)$ is a diagonal matrix whose $i$th diagonal entry is the transmission of the vertex $i$ in $G$. In this paper, first, upper and lower bounds for the spectral radius of a nonnegative matrix are constructed. Applying this result, upper and lower bounds for the distance and distance signless Laplacian spectral radius of graphs are given, and the extremal graphs for these bounds are obtained. Also, upper bounds for the modulus of all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius of graphs are given. These bounds are probably first of their kind as the authors do not find in the literature any bound for these eigenvalues. Finally, for some classes of graphs, it is shown that all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius lie in the smallest Ger\^sgorin disc of the distance (respectively, distance signless Laplacian) matrix.

scholarly journals Upper bounds on Q-spectral radius of book-free and/or $K_{s,t}$-free graphs

2017 ◽  
Vol 32 ◽  
pp. 447-453
Author(s):  
Qi Kong ◽  
Ligong Wang
Keyword(s):  
Spectral Radius ◽  
Regular Graph ◽  
Maximum Degree ◽  
Strongly Regular Graph ◽  
Upper Bounds ◽  
Free Graph ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian ◽  
Strongly Regular

In this paper, we prove two results about the signless Laplacian spectral radius $q(G)$ of a graph $G$ of order $n$ with maximum degree $\Delta$. Let $B_{n}=K_{2}+\overline{K_{n}}$ denote a book, i.e., the graph $B_{n}$ consists of $n$ triangles sharing an edge. The results are the following: (1) Let $1< k\leq l< \Delta < n$ and $G$ be a connected \{$B_{k+1},K_{2,l+1}$\}-free graph of order $n$ with maximum degree $\Delta$. Then $$\displaystyle q(G)\leq \frac{1}{4}[3\Delta+k-2l+1+\sqrt{(3\Delta+k-2l+1)^{2}+16l(\Delta+n-1)}$$ with equality if and only if $G$ is a strongly regular graph with parameters ($\Delta$, $k$, $l$). (2) Let $s\geq t\geq 3$, and let $G$ be a connected $K_{s,t}$-free graph of order $n$ $(n\geq s+t)$. Then $$q(G)\leq n+(s-t+1)^{1/t}n^{1-1/t}+(t-1)(n-1)^{1-3/t}+t-3.$$

scholarly journals Sharp Lower Bounds on the Spectral Radius of Uniform Hypergraphs Concerning Degrees

10.37236/6644 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Liying Kang ◽  
Lele Liu ◽  
Erfang Shan
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Linear Algebra ◽  
Lower Bounds ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Signless Laplacian Tensor ◽  
Signless Laplacian ◽  
Vertex Degrees

Let $\mathcal{A}(H)$ and $\mathcal{Q}(H)$ be the adjacency tensor and signless Laplacian tensor of an $r$-uniform hypergraph $H$. Denote by $\rho(H)$ and $\rho(\mathcal{Q}(H))$ the spectral radii of $\mathcal{A}(H)$ and $\mathcal{Q}(H)$, respectively. In this paper we present a  lower bound on $\rho(H)$ in terms of vertex degrees and we characterize the extremal hypergraphs attaining the bound, which solves a problem posed by Nikiforov [Analytic methods for uniform hypergraphs, Linear Algebra Appl. 457 (2014) 455–535]. Also, we prove a lower bound on $\rho(\mathcal{Q}(H))$ concerning degrees and give a characterization of the extremal hypergraphs attaining the bound.

SPECTRAL RADIUS AND AVERAGE 2-DEGREE SEQUENCE OF A GRAPH

2014 ◽  
Vol 06 (02) ◽  
pp. 1450029 ◽  
Author(s):  
YU-PEI HUANG ◽  
CHIH-WEN WENG
Keyword(s):  
Spectral Radius ◽  
Maximum Degree ◽  
Degree Sequence ◽  
Discrete Math ◽  
Upper Bounds ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Degree Of A Vertex ◽  
Signless Laplacian ◽  
Simple Connected Graph

In a simple connected graph, the average 2-degree of a vertex is the average degree of its neighbors. With the average 2-degree sequence and the maximum degree ratio of adjacent vertices, we present a sharp upper bound of the spectral radius of the adjacency matrix of a graph, which improves a result in [Y. H. Chen, R. Y. Pan and X. D. Zhang, Two sharp upper bounds for the signless Laplacian spectral radius of graphs, Discrete Math. Algorithms Appl.3(2) (2011) 185–191].

scholarly journals On Distance Signless Laplacian Spectral Radius and Distance Signless Laplacian Energy

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 792
Author(s):  
Luis Medina ◽  
Hans Nina ◽  
Macarena Trigo
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Connected Graph ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian Spectral Radius ◽  
Laplacian Energy ◽  
Signless Laplacian

In this article, we find sharp lower bounds for the spectral radius of the distance signless Laplacian matrix of a simple undirected connected graph and we apply these results to obtain sharp upper bounds for the distance signless Laplacian energy graph. The graphs for which those bounds are attained are characterized.

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