scholarly journals Signless Laplacian determinations of some graphs with independent edges

2018 ◽  
Vol 10 (1) ◽  
pp. 185-196 ◽  
Author(s):  
R. Sharafdini ◽  
A.Z. Abdian
Keyword(s):  
Natural Number ◽  
Complete Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Laplacian Matrix ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Laplacian Spectra ◽  
Signless Laplacian ◽  
Degree Matrix

Let $G$ be a simple undirected graph. Then the signless Laplacian matrix of $G$ is defined as $D_G + A_G$ in which $D_G$ and $A_G$ denote the degree matrix and the adjacency matrix of $G$, respectively. The graph $G$ is said to be determined by its signless Laplacian spectrum (DQS, for short), if any graph having the same signless Laplacian spectrum as $G$ is isomorphic to $G$. We show that $G\sqcup rK_2$ is determined by its signless Laplacian spectra under certain conditions, where $r$ and $K_2$ denote a natural number and the complete graph on two vertices, respectively. Applying these results, some DQS graphs with independent edges are obtained.

The distance Laplacian and distance signless Laplacian spectrum of the subdivision-vertex join and subdivision-edge join of two regular graphs

2019 ◽  
Vol 11 (05) ◽  
pp. 1950053
Author(s):  
Deena C. Scaria ◽  
G. Indulal
Keyword(s):  
Regular Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Subdivision Graph ◽  
Laplacian Spectra ◽  
Signless Laplacian ◽  
Vertex Disjoint ◽  
Jahangir Graph

Let [Formula: see text] be a connected graph with a distance matrix [Formula: see text]. Let [Formula: see text] and [Formula: see text] be, respectively, the distance Laplacian matrix and the distance signless Laplacian matrix of graph [Formula: see text], where [Formula: see text] denotes the diagonal matrix of the vertex transmissions in [Formula: see text]. The eigenvalues of [Formula: see text] and [Formula: see text] constitute the distance Laplacian spectrum and distance signless Laplacian spectrum, respectively. The subdivision graph [Formula: see text] of a graph [Formula: see text] is obtained by inserting a new vertex into every edge of [Formula: see text]. We denote the set of such new vertices by [Formula: see text]. The subdivision-vertex join of two vertex disjoint graphs [Formula: see text] and [Formula: see text] denoted by [Formula: see text], is the graph obtained from [Formula: see text] and [Formula: see text] by joining each vertex of [Formula: see text] with every vertex of [Formula: see text]. The subdivision-edge join of two vertex disjoint graphs [Formula: see text] and [Formula: see text] denoted by [Formula: see text], is the graph obtained from [Formula: see text] and [Formula: see text] by joining each vertex of [Formula: see text] with every vertex of [Formula: see text]. In this paper, we determine the distance Laplacian and distance signless Laplacian spectra of subdivision-vertex join and subdivision-edge join of a connected regular graph with an arbitrary regular graph in terms of their eigenvalues. As an application we exhibit some infinite families of cospectral graphs and find the respective spectra of the Jahangir graph [Formula: see text].

scholarly journals On the Adjacency, Laplacian, and Signless Laplacian Spectrum of Coalescence of Complete Graphs

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
S. R. Jog ◽  
Raju Kotambari
Keyword(s):  
Line Graph ◽  
Laplacian Matrix ◽  
Spectral Graph Theory ◽  
Complete Graphs ◽  
Laplacian Spectrum ◽  
Spectral Graph ◽  
Signless Laplacian Spectrum ◽  
Subdivision Graph ◽  
Laplacian Energy ◽  
Signless Laplacian

Coalescence as one of the operations on a pair of graphs is significant due to its simple form of chromatic polynomial. The adjacency matrix, Laplacian matrix, and signless Laplacian matrix are common matrices usually considered for discussion under spectral graph theory. In this paper, we compute adjacency, Laplacian, and signless Laplacian energy (Qenergy) of coalescence of pair of complete graphs. Also, as an application, we obtain the adjacency energy of subdivision graph and line graph of coalescence from itsQenergy.

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 A note on super integral rings

2019 ◽  
Vol 38 (4) ◽  
pp. 213-218 ◽  
Author(s):  
Rajat Kanti Nath
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Commuting Graph ◽  
Signless Laplacian ◽  
Vertex Set

Let $R$ be a nite non-commutative ring with center $Z(R)$. The commuting graph of $R$, denoted by $\Gamma_R$, is a simple undirected graph whose vertex set is $R\setminus Z(R)$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy = yx$. Let$\Spec(\Gamma_R),  \L-Spec(\GammaR)$ and $\Q-Spec(\GammaR)$ denote the spectrum, Laplacian spectrum and signless Laplacian spectrum of  $\Gamma_R$ respectively. A nite non-commutative ring $R$ is called super integral if $\Spec(\Gamma_R), \L-Spec(Gamma_R)$ and $\Q-Spec(\Gamma_R)$ contain only integers. In this paper, we obtain several classes of super integral rings.

Spectrum of graphs obtained by operations

2018 ◽  
Vol 13 (02) ◽  
pp. 2050045
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Somnath Paul
Keyword(s):  
Distance Matrix ◽  
Laplacian Matrix ◽  
Regular Graphs ◽  
Lexicographic Product ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian ◽  
Adjacency Spectrum ◽  
Simple Connected Graph ◽  
Equienergetic Graphs

The distance signless Laplacian matrix of a simple connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix whose main diagonal entries are the vertex transmissions in [Formula: see text]. In this paper, we first determine the distance signless Laplacian spectrum of the graphs obtained by generalization of the join and lexicographic product graph operations (namely joined union) in terms of their adjacency spectrum and the eigenvalues of an auxiliary matrix, determined by the graph [Formula: see text]. As an application, we show that new pairs of auxiliary equienergetic graphs can be constructed by joined union of regular graphs.

The signless Laplacian spectrum of rooted product of graphs

2018 ◽  
Vol 10 (06) ◽  
pp. 1850076 ◽  
Author(s):  
Maryam Maghsoudi ◽  
Abbas Heydari
Keyword(s):  
Characteristic Polynomial ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Product Formula ◽  
Laplacian Spectrum ◽  
Root Vertex ◽  
Signless Laplacian Spectrum ◽  
Laplacian Energy ◽  
Signless Laplacian ◽  
Product Of Graphs

Let [Formula: see text] be a simple graph with [Formula: see text] vertices and [Formula: see text] be a sequence of [Formula: see text] rooted graphs [Formula: see text]. The rooted product [Formula: see text], of [Formula: see text] by [Formula: see text] is constructed by identifying the root vertex of [Formula: see text] with the [Formula: see text]th vertex of [Formula: see text] for [Formula: see text]. In this paper, we introduce a method for computation of the characteristic polynomial of the signless Laplacian matrix of [Formula: see text]. Then the signless Laplacian spectrum and the signless Laplacian energy of some graphs will be computed.

The spectra of a new join of graphs

2018 ◽  
Vol 10 (06) ◽  
pp. 1850074 ◽  
Author(s):  
Somnath Paul
Keyword(s):  
Laplacian Spectrum ◽  
Cospectral Graphs ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian ◽  
Join Of Graphs ◽  
Disjoint Sets

Let [Formula: see text] and [Formula: see text] be three graphs on disjoint sets of vertices and [Formula: see text] has [Formula: see text] edges. Let [Formula: see text] be the graph obtained from [Formula: see text] and [Formula: see text] in the following way: (1) Delete all the edges of [Formula: see text] and consider [Formula: see text] disjoint copies of [Formula: see text]. (2) Join each vertex of the [Formula: see text]th copy of [Formula: see text] to the end vertices of the [Formula: see text]th edge of [Formula: see text]. Let [Formula: see text] be the graph obtained from [Formula: see text] by joining each vertex of [Formula: see text] with each vertex of [Formula: see text] In this paper, we determine the adjacency (respectively, Laplacian, signless Laplacian) spectrum of [Formula: see text] in terms of those of [Formula: see text] and [Formula: see text] As an application, we construct infinite pairs of cospectral graphs.

scholarly journals Bounds for the Generalized Distance Eigenvalues of a Graph

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1529 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal Ahmad Ganie ◽  
Yilun Shang
Keyword(s):  
Spectral Radius ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Extremal Graphs ◽  
Special Cases ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Generalized Distance

Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian, D Q ( G ) be the distance signless Laplacian, and T r ( G ) be the diagonal matrix of the vertex transmissions, respectively. Furthermore, we denote by D α ( G ) the generalized distance matrix, i.e., D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where α ∈ [ 0 , 1 ] . In this paper, we establish some new sharp bounds for the generalized distance spectral radius of G, making use of some graph parameters like the order n, the diameter, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α , improving some bounds recently given in the literature. We also characterize the extremal graphs attaining these bounds. As an special cases of our results, we will be able to cover some of the bounds recently given in the literature for the case of distance matrix and distance signless Laplacian matrix. We also obtain new bounds for the k-th generalized distance eigenvalue.

scholarly journals Bounds of Laplacian Energy of a Hypercube Graph

2018 ◽  
Vol 7 (4.10) ◽  
pp. 582
Author(s):  
K. Ameenal Bibi ◽  
B. Vijayalakshmi ◽  
R. Jothilakshmi
Keyword(s):  
Lower Bounds ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Positive Semidefinite ◽  
Laplacian Energy ◽  
Eigen Values ◽  
Degree Matrix

Let  Qn denote  the n – dimensional  hypercube  with  order   2n and  size n2n-1. The  Laplacian  L  is defined  by  L = D  where D is  the  degree  matrix and  A is  the  adjacency  matrix  with  zero  diagonal  entries.  The  Laplacian  is a  symmetric  positive  semidefinite.  Let  µ1 ≥ µ2 ≥ ....µn-1 ≥ µn = 0 be the eigen values of  the Laplacian matrix.  The  Laplacian  energy is defined as  LE(G) = . In  this  paper, we  defined  Laplacian  energy  of  a  Hypercube  graph  and  also attained  the  lower  bounds.   

scholarly journals Properties of the characteristic polynomials and spectrum of Pn and Cn

2016 ◽  
Vol 5 (2) ◽  
pp. 132
Author(s):  
Essam El Seidy ◽  
Salah Eldin Hussein ◽  
Atef Mohamed
Keyword(s):  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Characteristic Polynomials ◽  
Signless Laplacian Matrix ◽  
Path Graph ◽  
Signless Laplacian ◽  
Vertex Set ◽  
Cycle Graph

We consider a finite undirected and connected simple graph  with vertex set  and edge set .We calculated the general formulas of the spectra of a cycle graph and path graph. In this discussion we are interested in the adjacency matrix, Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, and seidel adjacency matrix.

Sign in / Sign up

Export Citation Format

Share Document