On (distance) signless Laplacian spectra of graphs

2021 ◽  
Author(s):  
B. R. Rakshith ◽  
Kinkar Chandra Das ◽  
M. A. Sriraj
Keyword(s):  
Spectra Of Graphs ◽  
Laplacian Spectra ◽  
Signless Laplacian

scholarly journals On the sum of signless Laplacian spectra of graphs

2019 ◽  
Vol 11 (2) ◽  
pp. 407-417 ◽  
Author(s):  
S. Pirzada ◽  
H.A. Ganie ◽  
A.M. Alghamdi
Keyword(s):  
Clique Number ◽  
Laplacian Spectrum ◽  
Laplacian Eigenvalues ◽  
Signless Laplacian Spectrum ◽  
Spectra Of Graphs ◽  
Laplacian Spectra ◽  
Vertex Covering ◽  
Signless Laplacian ◽  
Vertex Set ◽  
Large Families

For a simple graph $G(V,E)$ with $n$ vertices, $m$ edges, vertex set $V(G)=\{v_1, v_2, \dots, v_n\}$ and edge set $E(G)=\{e_1, e_2,\dots, e_m\}$, the adjacency matrix $A=(a_{ij})$ of $G$ is a $(0, 1)$-square matrix of order $n$ whose $(i,j)$-entry is equal to 1 if $v_i$ is adjacent to $v_j$ and equal to 0, otherwise. Let $D(G)={diag}(d_1, d_2, \dots, d_n)$ be the diagonal matrix associated to $G$, where $d_i=\deg(v_i),$ for all $i\in \{1,2,\dots,n\}$. The matrices $L(G)=D(G)-A(G)$ and $Q(G)=D(G)+A(G)$ are respectively called the Laplacian and the signless Laplacian matrices and their spectra (eigenvalues) are respectively called the Laplacian spectrum ($L$-spectrum) and the signless Laplacian spectrum ($Q$-spectrum) of the graph $G$. If $0=\mu_n\leq\mu_{n-1}\leq\cdots\leq\mu_1$ are the Laplacian eigenvalues of $G$, Brouwer conjectured that the sum of $k$ largest Laplacian eigenvalues $S_{k}(G)$ satisfies $S_{k}(G)=\sum\limits_{i=1}^{k}\mu_i\leq m+{k+1 \choose 2}$ and this conjecture is still open. If $q_1,q_2, \dots, q_n$ are the signless Laplacian eigenvalues of $G$, for $1\leq k\leq n$, let $S^{+}_{k}(G)=\sum_{i=1}^{k}q_i$ be the sum of $k$ largest signless Laplacian eigenvalues of $G$. Analogous to Brouwer's conjecture, Ashraf et al. conjectured that $S^{+}_{k}(G)\leq m+{k+1 \choose 2}$, for all $1\leq k\leq n$. This conjecture has been verified in affirmative for some classes of graphs. We obtain the upper bounds for $S^{+}_{k}(G)$ in terms of the clique number $\omega$, the vertex covering number $\tau$ and the diameter of the graph $G$. Finally, we show that the conjecture holds for large families of graphs.

scholarly journals Results on Laplacian spectra of graphs with pockets

2018 ◽  
Vol 15 (1) ◽  
pp. 79-87 ◽  
Author(s):  
Sasmita Barik ◽  
Gopinath Sahoo
Keyword(s):  
Spectra Of Graphs ◽  
Laplacian Spectra

The spectral characterizations of the connected multicone graphs Kw ▽ LHS and Kw ▽ LGQ(3,9)

2018 ◽  
Vol 10 (02) ◽  
pp. 1850019 ◽  
Author(s):  
Ali Zeydi Abdian ◽  
S. Morteza Mirafzal
Keyword(s):  
Optimization Problems ◽  
Spectral Graph Theory ◽  
Combinatorial Optimization Problems ◽  
Spectral Graph ◽  
The Past ◽  
Laplacian Spectra ◽  
Signless Laplacian ◽  
Multicone Graph ◽  
Spectral Characterizations ◽  
Sims Graph

In the past decades, graphs that are determined by their spectrum have received much more and more attention, since they have been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. An important part of spectral graph theory is devoted to determining whether given graphs or classes of graphs are determined by their spectra or not. So, finding and introducing any class of graphs which are determined by their spectra can be an interesting and important problem. The main aim of this study is to characterize two classes of multicone graphs which are determined by their adjacency, Laplacian and signless Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let [Formula: see text] denote a complete graph on [Formula: see text] vertices. In the paper, we show that multicone graphs [Formula: see text] and [Formula: see text] are determined by both their adjacency spectra and their Laplacian spectra, where [Formula: see text] and [Formula: see text] denote the Local Higman–Sims graph and the Local [Formula: see text] graph, respectively. In addition, we prove that these multicone graphs are determined by their signless Laplacian spectra.

An edge-grafting theorem on Laplacian spectra of graphs and its application

2011 ◽  
Vol 59 (3) ◽  
pp. 303-315 ◽  
Author(s):  
Mingqing Zhai ◽  
Ruifang Liu ◽  
Jinlong Shu
Keyword(s):  
Spectra Of Graphs ◽  
Laplacian Spectra

scholarly journals Spectra of Graphs Resulting from Various Graph Operations and Products: a Survey

2018 ◽  
Vol 6 (1) ◽  
pp. 323-342 ◽  
Author(s):  
S. Barik ◽  
D. Kalita ◽  
S. Pati ◽  
G. Sahoo
Keyword(s):  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Explicit Description ◽  
Graph Products ◽  
Graph Operations ◽  
Signless Laplacian Matrix ◽  
Spectra Of Graphs ◽  
Signless Laplacian

AbstractLet G be a graph on n vertices and A(G), L(G), and |L|(G) be the adjacency matrix, Laplacian matrix and signless Laplacian matrix of G, respectively. The paper is essentially a survey of known results about the spectra of the adjacency, Laplacian and signless Laplacian matrix of graphs resulting from various graph operations with special emphasis on corona and graph products. In most cases, we have described the eigenvalues of the resulting graphs along with an explicit description of the structure of the corresponding eigenvectors.

scholarly journals Graphs determined by signless Laplacian spectra

2020 ◽  
Vol 17 (1) ◽  
pp. 45-50 ◽  
Author(s):  
Ali Zeydi Abdian ◽  
Afshin Behmaram ◽  
Gholam Hossein Fath-Tabar
Keyword(s):  
Laplacian Spectra ◽  
Signless Laplacian
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