scholarly journals On the Spectra of Commuting and Non Commuting Graph on Dihedral Group

CAUCHY ◽  
2017 ◽  
Vol 4 (4) ◽  
pp. 176 ◽  
Author(s):  
Abdussakir Abdussakir ◽  
Rivatul Ridho Elvierayani ◽  
Muflihatun Nafisah
Keyword(s):  
Dihedral Group ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Commuting Graph ◽  
Signless Laplacian ◽  
Adjacency Spectrum

Study about spectra of graph has became interesting work as well as study about commuting and non commuting graph of a group or a ring. But the study about spectra of commuting and non commuting graph of dihedral group has not been done yet. In this paper, we investigate adjacency spectrum, Laplacian spectrum, signless Laplacian spectrum, and detour spectrum of commuting and non commuting graph of dihedral group <em>D</em><sub>2<em>n</em></sub>

scholarly journals Spectral Characterizations of Dumbbell Graphs

10.37236/314 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Jianfeng Wang ◽  
Francesco Belardo ◽  
Qiongxiang Huang ◽  
Enzo M. Li Marzi
Keyword(s):  
Spectral Characterization ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Disjoint Cycles ◽  
Signless Laplacian ◽  
Adjacency Spectrum ◽  
Unique Graph ◽  
Vertex Disjoint ◽  
Spectral Characterizations

A dumbbell graph, denoted by $D_{a,b,c}$, is a bicyclic graph consisting of two vertex-disjoint cycles $C_a$, $C_b$ and a path $P_{c+3}$ ($c \geq -1$) joining them having only its end-vertices in common with the two cycles. In this paper, we study the spectral characterization w.r.t. the adjacency spectrum of $D_{a,b,0}$ (without cycles $C_4$) with $\gcd(a,b)\geq 3$, and we complete the research started in [J.F. Wang et al., A note on the spectral characterization of dumbbell graphs, Linear Algebra Appl. 431 (2009) 1707–1714]. In particular we show that $D_{a,b,0}$ with $3 \leq \gcd(a,b) < a$ or $\gcd(a,b)=a$ and $b\neq 3a$ is determined by the spectrum. For $b=3a$, we determine the unique graph cospectral with $D_{a,3a,0}$. Furthermore we give the spectral characterization w.r.t. the signless Laplacian spectrum of all dumbbell 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.

scholarly journals Spectra of Subdivision Vertex-Edge Join of Three Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 171 ◽  
Author(s):  
Fei Wen ◽  
You Zhang ◽  
Muchun Li
Keyword(s):  
Spanning Trees ◽  
Regular Graphs ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian ◽  
Adjacency Spectrum ◽  
Graph Operation ◽  
Arbitrary Graphs ◽  
Number Of Spanning Trees

In this paper, we introduce a new graph operation called subdivision vertex-edge join (denoted by G 1 S ▹ ( G 2 V ∪ G 3 E ) for short), and then the adjacency spectrum, the Laplacian spectrum and the signless Laplacian spectrum of G 1 S ▹ ( G 2 V ∪ G 3 E ) are respectively determined in terms of the corresponding spectra for a regular graph G 1 and two arbitrary graphs G 2 and G 3 . All the above can be viewed as the generalizations of the main results in [X. Liu, Z. Zhang, Bull. Malays. Math. Sci. Soc., 2017:1–17]. Furthermore, we also determine the normalized Laplacian spectrum of G 1 S ▹ ( G 2 V ∪ G 3 E ) whenever G i are regular graphs for each index i = 1 , 2 , 3 . As applications, we construct infinitely many pairs of A-cospectral mates, L-cospectral mates, Q-cospectral mates and L -cospectral mates. Finally, we give the number of spanning trees, the (degree-)Kirchhoff index and the Kemeny’s constant of G 1 S ▹ ( G 2 V ∪ G 3 E ) , respectively.

THE DISTANCE RELATED SPECTRA OF SOME SUBDIVISION RELATED GRAPHS

2021 ◽  
Vol 3 (1) ◽  
pp. 22-36
Author(s):  
I. Gopalapillai ◽  
D.C. Scaria
Keyword(s):  
Connected Graph ◽  
Distance Matrix ◽  
Transmission Matrix ◽  
Laplacian Spectrum ◽  
Laplacian Matrices ◽  
Signless Laplacian Spectrum ◽  
Subdivision Graph ◽  
Signless Laplacian ◽  
Analytic Expressions ◽  
Adjacency Spectrum

Let $G$ be a connected graph with a distance matrix $D$. The distance eigenvalues of $G$ are the eigenvalues of $D$, and the distance energy $E_D(G)$ is the sum of its absolute values. The transmission $Tr(v)$ of a vertex $v$ is the sum of the distances from $v$ to all other vertices in $G$. The transmission matrix $Tr(G)$ of $G$ is a diagonal matrix with diagonal entries equal to the transmissions of vertices. The matrices $D^L(G)= Tr(G)-D(G)$ and $D^Q(G)=Tr(G)+D(G)$ are, respectively, the Distance Laplacian and the Distance Signless Laplacian matrices of $G$. The eigenvalues of $D^L(G)$ ( $D^Q(G)$) constitute the Distance Laplacian spectrum ( Distance Signless Laplacian spectrum ). The subdivision graph $S(G)$ of $G$ is obtained by inserting a new vertex into every edge of $G$. We describe here the Distance Spectrum, Distance Laplacian spectrum and Distance Signless Laplacian spectrum of some types of subdivision related graphs of a regular graph in the terms of its adjacency spectrum. We also derive analytic expressions for the distance energy of $\bar{S}(C_p)$, partial complement of the subdivision of a cycle $C_p$ and that of $\overline {S\left( {C_p }\right)}$, complement of the even cycle $C_{2p}$.

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 On Q-integral (3,s)-semiregular bipartite graphs

2010 ◽  
Vol 4 (1) ◽  
pp. 167-174 ◽  
Author(s):  
Slobodan Simic ◽  
Zoran Stanic
Keyword(s):  
Bipartite Graphs ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian

A graph is called Q-integral if its signless Laplacian spectrum consists entirely of integers. We establish some general results regarding signless Laplacians of semiregular bipartite graphs. Especially, we consider those semiregular bipartite graphs with integral signless Laplacian spectrum. In some particular cases we determine the possible Q-spectra and consider the corresponding graphs.

On the signless Laplacian spectral determination of the join of regular graphs

2014 ◽  
Vol 06 (04) ◽  
pp. 1450050
Author(s):  
Lizhen Xu ◽  
Changxiang He
Keyword(s):  
Regular Graph ◽  
Spectral Determination ◽  
Regular Graphs ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian

Let G be an r-regular graph with order n, and G ∨ H be the graph obtained by joining each vertex of G to each vertex of H. In this paper, we prove that G ∨ K2is determined by its signless Laplacian spectrum for r = 1, n - 2. For r = n - 3, we show that G ∨ K2is determined by its signless Laplacian spectrum if and only if the complement of G has no triangles.

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.

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