Distance Laplacian spectra of joined union of graphs

2021 ◽  
pp. 2250039
Author(s):  
Somnath Paul
Keyword(s):  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
Lexicographic Product ◽  
Combinatorial Laplacian ◽  
Laplacian Spectra ◽  
Weighted Laplacian ◽  
Simple Connected Graph ◽  
Product Of Graphs

The distance 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 determine the distance Laplacian spectra of the graphs obtained by generalization of the join and lexicographic product of graphs (namely joined union). It is shown that the distance Laplacian spectra of these graphs not only depend on the distance Laplacian spectra of the participating graphs but also depend on the spectrum of another matrix of vertex-weighted Laplacian kind (analogous to the definition given by Chung and Langlands [A combinatorial Laplacian with vertex weights, J. Combin. Theory Ser. A 75 (1996) 316–327]).

scholarly journals On the Ger\v{s}gorin disks of distance matrices of graphs

2021 ◽  
Vol 37 ◽  
pp. 709-717
Author(s):  
Mustapha Aouchiche ◽  
Bilal A. Rather ◽  
Issmail El Hallaoui
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Infinite Family ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Negative Answer ◽  
Diagonal Matrix ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian ◽  
Simple Connected Graph

For a simple connected graph $G$, let $D(G)$, $Tr(G)$, $D^{L}(G)=Tr(G)-D(G)$, and $D^{Q}(G)=Tr(G)+D(G)$ be the distance matrix, the diagonal matrix of the vertex transmissions, the distance Laplacian matrix, and the distance signless Laplacian matrix of $G$, respectively. Atik and Panigrahi [2] suggested the study of the problem: Whether all eigenvalues, except the spectral radius, of $ D(G) $ and $ D^{Q}(G) $ lie in the smallest Ger\v{s}gorin disk? In this paper, we provide a negative answer by constructing an infinite family of counterexamples.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals Minimum covering reciprocal distance signless Laplacian energy of graphs

2018 ◽  
Vol 10 (2) ◽  
pp. 218-240
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Ebrahim Hashemi ◽  
Yaser Alizadeh
Keyword(s):  
Connected Graph ◽  
Vertex Cover ◽  
Cartesian Product ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Double Cover ◽  
Laplacian Energy ◽  
Signless Laplacian ◽  
The Matrix ◽  
Simple Connected Graph

Abstract Let G be a simple connected graph. The reciprocal transmission Tr′G(ν) of a vertex ν is defined as $${\rm{Tr}}_{\rm{G}}^\prime ({\rm{\nu }}) = \sum\limits_{{\rm{u}} \in {\rm{V}}(G)} {{1 \over {{{\rm{d}}_{\rm{G}}}(u,{\rm{\nu }})}}{\rm{u}} \ne {\rm{\nu }}.} $$ The reciprocal distance signless Laplacian (briefly RDSL) matrix of a connected graph G is defined as RQ(G)= diag(Tr′ (G)) + RD(G), where RD(G) is the Harary matrix (reciprocal distance matrix) of G and diag(Tr′ (G)) is the diagonal matrix of the vertex reciprocal transmissions in G. In this paper, we investigate the RDSL spectrum of some classes of graphs that are arisen from graph operations such as cartesian product, extended double cover product and InduBala product. We introduce minimum covering reciprocal distance signless Laplacian matrix (or briey MCRDSL matrix) of G as the square matrix of order n, RQC(G) := (qi;j), $${{\rm{q}}_{{\rm{ij}}}} = \left\{ {\matrix{ {1 + {\rm{Tr}}\prime ({{\rm{\nu }}_{\rm{i}}})} & {{\rm{if}}} & {{\rm{i = j}}} & {{\rm{and}}} & {{{\rm{\nu }}_{\rm{i}}} \in {\rm{C}}} \cr {{\rm{Tr}}\prime ({{\rm{\nu }}_{\rm{i}}})} & {{\rm{if}}} & {{\rm{i = j}}} & {{\rm{and}}} & {{{\rm{\nu }}_{\rm{i}}} \notin {\rm{C}}} \cr {{1 \over {{\rm{d(}}{{\rm{\nu }}_{\rm{i}}},{{\rm{\nu }}_{\rm{j}}})}}} & {{\rm{otherwise}}} & {} & {} & {} \cr } } \right.$$ where C is a minimum vertex cover set of G. MCRDSL energy of a graph G is defined as sum of eigenvalues of RQC. Extremal graphs with respect to MCRDSL energy of graph are characterized. We also obtain some bounds on MCRDSL energy of a graph and MCRDSL spectral radius of 𝒢, which is the largest eigenvalue of the matrix RQC (G) of graphs.

scholarly journals On the Sum of Distance Laplacian Eigenvalues of Graphs

2021 ◽  
Vol 54 ◽  
Author(s):  
Shariefuddin Pirzada ◽  
Saleem Khan
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Wiener Index ◽  
Upper Bounds ◽  
Diagonal Matrix ◽  
Laplacian Eigenvalues ◽  
Eigenvalues Of Graphs

Let $G$ be a connected graph with $n$ vertices, $m$ edges and having diameter $d$. The distance Laplacian matrix $D^{L}$ is defined as $D^L=$Diag$(Tr)-D$, where Diag$(Tr)$ is the diagonal matrix of vertex transmissions and $D$ is the distance matrix of $G$. The distance Laplacian eigenvalues of $G$ are the eigenvalues of $D^{L}$ and are denoted by $\delta_{1}, ~\delta_{1},~\dots,\delta_{n}$. In this paper, we obtain (a) the upper bounds for the sum of $k$ largest and (b) the lower bounds for the sum of $k$ smallest non-zero, distance Laplacian eigenvalues of $G$ in terms of order $n$, diameter $d$ and Wiener index $W$ of $G$. We characterize the extremal cases of these bounds. As a consequence, we also obtain the bounds for the sum of the powers of the distance Laplacian eigenvalues of $G$.

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 The Distance Laplacian Spectral Radius of Clique Trees

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Xiaoling Zhang ◽  
Jiajia Zhou
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Minimum Distance ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
Laplacian Spectral Radius ◽  
Largest Eigenvalue ◽  
Clique Trees ◽  
The Largest Eigenvalue

The distance Laplacian matrix of a connected graph G is defined as ℒ G = Tr G − D G , where D G is the distance matrix of G and Tr G is the diagonal matrix of vertex transmissions of G . The largest eigenvalue of ℒ G is called the distance Laplacian spectral radius of G . In this paper, we determine the graphs with maximum and minimum distance Laplacian spectral radius among all clique trees with n vertices and k cliques. Moreover, we obtain n vertices and k cliques.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Joins, coronas and their vertex-edge Wiener polynomials

2016 ◽  
Vol 47 (2) ◽  
pp. 163-178
Author(s):  
Mahdieh Azari ◽  
Ali Iranmanesh
Keyword(s):  
Generating Function ◽  
Connected Graph ◽  
Wiener Index ◽  
First Derivative ◽  
Sum Of Distances ◽  
Simple Connected Graph ◽  
Product Of Graphs ◽  
Corona Product

The vertex-edge Wiener index of a simple connected graph $G$ is defined as the sum of distances between vertices and edges of $G$. The vertex-edge Wiener polynomial of $G$ is a generating function whose first derivative is a $q-$analog of the vertex-edge Wiener index. Two possible distances $D_1(u, e|G)$ and $D_2(u, e|G)$ between a vertex $u$ and an edge $e$ of $G$ can be considered and corresponding to them, the first and second vertex-edge Wiener indices of $G$, and the first and second vertex-edge Wiener polynomials of $G$ are introduced. In this paper, we study the behavior of these indices and polynomials under the join and corona product of graphs. Results are applied for some classes of graphs such as suspensions, bottlenecks, and thorny graphs.

scholarly journals Identifying Codes of Lexicographic Product of Graphs

10.37236/2974 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Min Feng ◽  
Min Xu ◽  
Kaishun Wang
Keyword(s):  
Connected Graph ◽  
Necessary Condition ◽  
Lexicographic Product ◽  
Arbitrary Graph ◽  
Minimum Cardinality ◽  
Identifying Codes ◽  
Two Parameters ◽  
Sufficient And Necessary Condition ◽  
Product Of Graphs

Let $G$ be a connected graph and $H$ be an arbitrary graph. In this paper, we study the identifying codes of the lexicographic product $G[H]$ of $G$ and $H$. We first introduce two parameters of $H$, which are closely related to identifying codes of $H$. Then we provide the sufficient and necessary condition for $G[H]$ to be identifiable. Finally, if $G[H]$ is identifiable, we determine the minimum cardinality of identifying codes of $G[H]$ in terms of the order of $G$ and these two parameters of $H$.

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

2018 ◽  
Vol 34 ◽  
pp. 459-471 ◽  
Author(s):  
Shuting Liu ◽  
Jinlong Shu ◽  
Jie Xue
Keyword(s):  
Spectral Radius ◽  
Regular Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Wiener Index ◽  
Regular Graphs ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Sum Of Distances

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.

Sign in / Sign up

Export Citation Format

Share Document