scholarly journals Constructions for cospectral graphs for the normalized Laplacian matrix and distance matrix

2017 ◽  
Author(s):  
Kristin Elizabeth Heysse
Keyword(s):  
Distance Matrix ◽  
Laplacian Matrix ◽  
Cospectral Graphs

scholarly journals Graphs that are cospectral for the distance Laplacian

2020 ◽  
Vol 36 (36) ◽  
pp. 334-351 ◽  
Author(s):  
Boris Brimkov ◽  
Ken Duna ◽  
Leslie Hogben ◽  
Kate Lorenzen ◽  
Carolyn Reinhart ◽  
...  
Keyword(s):  
Characteristic Polynomial ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
Circulant Graphs ◽  
Cospectral Graphs ◽  
The Matrix ◽  
Strongly Regular ◽  
The Absolute

The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is $\mathcal{D}^L (G)=T(G)-\mathcal{D} (G)$, where $T(G)$ is the diagonal matrix of row sums of $\mathcal{D}(G)$. Several general methods are established for producing $\mathcal{D}^L$-cospectral graphs that can be used to construct infinite families. Examples are provided to show that various properties are not preserved by $\mathcal{D}^L$-cospectrality, including examples of $\mathcal{D}^L$-cospectral strongly regular and circulant graphs. It is established that the absolute values of coefficients of the distance Laplacian characteristic polynomial are decreasing, i.e., $|\delta^L_{1}|\geq \cdots \geq |\delta^L_{n}|$, where $\delta^L_{k}$ is the coefficient of $x^k$.  

scholarly journals Cospectral constructions for several graph matrices using cousin vertices

2021 ◽  
Vol 10 (1) ◽  
pp. 9-22
Author(s):  
Kate Lorenzen
Keyword(s):  
Adjacency Matrix ◽  
Structural Information ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Cospectral Graphs ◽  
Combinatorial Laplacian ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian

Abstract Graphs can be associated with a matrix according to some rule and we can find the spectrum of a graph with respect to that matrix. Two graphs are cospectral if they have the same spectrum. Constructions of cospectral graphs help us establish patterns about structural information not preserved by the spectrum. We generalize a construction for cospectral graphs previously given for the distance Laplacian matrix to a larger family of graphs. In addition, we show that with appropriate assumptions this generalized construction extends to the adjacency matrix, combinatorial Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, and distance matrix. We conclude by enumerating the prevelance of this construction in small graphs for the adjacency matrix, combinatorial Laplacian matrix, and distance Laplacian matrix.

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 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 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.

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.

On the sum of the distance signless Laplacian eigenvalues of a graph and some inequalities involving them

2019 ◽  
Vol 12 (01) ◽  
pp. 2050006 ◽  
Author(s):  
A. Alhevaz ◽  
M. Baghipur ◽  
E. Hashemi ◽  
S. Paul
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Graph Invariants ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
Signless Laplacian ◽  
Eigenvalues Of Graphs

The distance signless Laplacian matrix of a 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 of vertex transmissions of [Formula: see text]. If [Formula: see text] are the distance signless Laplacian eigenvalues of a simple graph [Formula: see text] of order [Formula: see text] then we put forward the graph invariants [Formula: see text] and [Formula: see text] for the sum of [Formula: see text]-largest and the sum of [Formula: see text]-smallest distance signless Laplacian eigenvalues of a graph [Formula: see text], respectively. We obtain lower bounds for the invariants [Formula: see text] and [Formula: see text]. Then, we present some inequalities between the vertex transmissions, distance eigenvalues, distance Laplacian eigenvalues, and distance signless Laplacian eigenvalues of graphs. Finally, we give some new results and bounds for the distance signless Laplacian energy of graphs.

Further results on the distance signless Laplacian spectrum of graphs

2018 ◽  
Vol 11 (05) ◽  
pp. 1850066 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Ebrahim Hashemi
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian ◽  
Graph Parameters

The distance signless Laplacian matrix [Formula: see text] of a 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 entries are the vertex transmissions of [Formula: see text], and the spectral radius of a connected graph [Formula: see text] is the largest eigenvalue of [Formula: see text]. In this paper, first we obtain the [Formula: see text]-eigenvalues of the join of certain regular graphs. Next, we give some new bounds on the distance signless Laplacian spectral radius of a graph [Formula: see text] in terms of graph parameters and characterize the extremal graphs. Utilizing these results we present some upper and lower bounds on the distance signless Laplacian energy of a graph [Formula: see text].

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.

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