Distance Laplacian Eigenvalues Research Articles

Distance Laplacian eigenvalues of graphs, and chromatic and independence number

Distance Laplacian eigenvalues of graphs, and chromatic and independence number

Reciprocal distance Laplacian spectral properties double stars and their complements

Several matrices are associated with graphs in order to study their properties. In such a study, researchers are interested in the spectra of the matrix under consideration, therefore, the properties are called spectral properties, with reference to the matrix. One of the interesting and hard problems in the spectral study of graphs is to order the graphs based on some spectral graph invariant, like the spectral radius, the second smallest eigenvalue, the energy, etc. Due to hardness of this problem it has been considered in the literature for small classes of graphs. Here we continue this study and add some more classes of graphs which can be ordered on the basis of spectral graph invariants. In this article, we study spectral properties of trees of diameter three, called double stars, and their complements through their reciprocal distance Laplacian eigenvalues. We give ordering of these graphs based on their reciprocal distance Laplacian spectral radius, on their second smallest reciprocal distance Laplacian eigenvalue, and on their reciprocal distance Laplacian energy.

On graphs with distance Laplacian eigenvalues of multiplicity n−4

Let G be a connected simple graph with n vertices. The distance Laplacian matrix D L ( G ) is defined as D L ( G ) = Diag ( T r ) − D ( G ) , where Diag ( T r ) is the diagonal matrix of vertex transmissions and D ( G ) is the distance matrix of G. The eigenvalues of D L ( G ) are the distance Laplacian eigenvalues of G and are denoted by ∂ 1 L ( G ) ≥ ∂ 2 L ( G ) ≥ ⋯ ≥ ∂ n L ( G ) . The largest eigenvalue ∂ 1 L ( G ) is called the distance Laplacian spectral radius. Lu et al. (2017), Fernandes et al. (2018), and Ma et al. (2018) completely characterized the graphs having some distance Laplacian eigenvalue of multiplicity n − 3 . In this paper, we characterize the graphs having distance Laplacian spectral radius of multiplicity n − 4 together with one of the distance Laplacian eigenvalues as n of multiplicity either 3 or 2. Further, we completely determine the graphs for which the distance Laplacian eigenvalue n is of multiplicity n − 4 .

Distance Laplacian spectral ordering of sun type graphs

Let G be a simple, connected graph of order n. Its distance Laplacian energy DLE(G) is given by DLE(G)=∑i=1n|ρiL−2W(G)n|, where ρ1L≥ρ2L≥⋯≥ρnL are the distance Laplacian eigenvalues and W(G) is the Wiener index of G. Distance Laplacian eigenvalues of sun and partial sun graphs have been characterized. We order the partial sun graphs by using their second largest distance Laplacian eigenvalue. Moreover, the distance Laplacian energy of sun and partial sun graphs have been derived in this paper. These graphs are also ordered by using their distance Laplacian energies.

On graphs with a few distinct reciprocal distance Laplacian eigenvalues

<abstract><p>For a $ \nu $-vertex connected graph $ \Gamma $, we consider the reciprocal distance Laplacian matrix defined as $ RD^L(\Gamma) = RT(\Gamma)-RD(\Gamma) $, i.e., $ RD^L(\Gamma) $ is the difference between the diagonal matrix of the reciprocal distance degrees $ RT(\Gamma) $ and the Harary matrix $ RD(\Gamma) $. In this article, we determine the graphs with exactly two distinct reciprocal distance Laplacian eigenvalues.We completely characterize the graph classes with a $ RD^L $ eigenvalue of multiplicity $ \nu-2 $. Moreover, we characterize families of graphs with reciprocal distance Laplacian eigenvalue whose multiplicity is $ \nu-3 $.</p></abstract>

On the distribution of eigenvalues of the reciprocal distance Laplacian matrix of graphs

The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G) = RT(G) ? RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Since RDL(G) is a real symmetric matrix, we denote its eigenvalues as ?1(RDL(G)) ? ?2(RDL(G)) ?...? ?n(RDL(G)). The largest eigenvalue ?1(RDL(G)) of RDL(G) is called the reciprocal distance Laplacian spectral radius. In this article, we prove that the multiplicity of n as a reciprocal distance Laplacian eigenvalue of RDL(G) is exactly one less than the number of components in the complement graph G of G. We show that the class of the complete bipartite graphs maximize the reciprocal distance Laplacian spectral radius among all the bipartite graphs with n vertices. Also, we show that the star graph Sn is the unique graph having the maximum reciprocal distance Laplacian spectral radius in the class of trees with n vertices. We determine the reciprocal distance Laplacian spectrum of several well known graphs. We prove that the complete graph Kn, Kn ? e, the star Sn, the complete balanced bipartite graph Kn/2 ,n/2 and the complete split graph CS(n, ?) are all determined from the RDL-spectrum.

Distance (Signless) Laplacian Eigenvalues of $k$-uniform Hypergraphs

The distance (signless) Laplacian eigenvalues of a connected hypergraph are the eigenvalues of its distance (signless) Laplacian matrix. For all $n$-vertex $k$-uniform hypertrees, we determine the $k$-uniform hypertree with minimum second largest distance (signless) Laplacian eigenvalue. For all $n$-vertex $k$-uniform unicyclic hypergraphs, we obtain the $k$-uniform unicyclic hypergraph with minimum largest distance (signless) Laplacian eigenvalue, and the $k$-uniform unicyclic hypergraph with minimum second largest distance Laplacian eigenvalue.

Eigenvalue localization and Geršgorin disc-related problems on distance and distance-related matrices of graphs

The distance, distance signless Laplacian and distance Laplacian matrix of a simple connected graph [Formula: see text] are denoted by [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] is the diagonal matrix of vertex transmission. Ger[Formula: see text]gorin discs for any [Formula: see text] square matrix [Formula: see text] are the discs [Formula: see text], where [Formula: see text]. The famous Ger[Formula: see text]gorin disc theorem says that all the eigenvalues of a square matrix lie in the union of the Ger[Formula: see text]gorin discs of that matrix. In this paper, some classes of graphs are studied for which the smallest Ger[Formula: see text]gorin disc contains every distance and distance signless Laplacian eigenvalues except the spectral radius of the corresponding matrix. For all connected graphs, a lower bound and for trees, an upper bound of every distance signless Laplacian eigenvalues except the spectral radius is given in this paper. These bounds are comparatively better than the existing bounds. By applying these bounds, we find some infinite classes of graphs for which the smallest Ger[Formula: see text]gorin disc contains every distance signless Laplacian eigenvalues except the spectral radius of the distance signless Laplacian matrix. For the distance Laplacian eigenvalues, we have given an upper bound and then find a condition for which the smallest Ger[Formula: see text]gorin disc contains every distance Laplacian eigenvalue of the distance Laplacian matrix. These results give partial answers from some questions that are raised in [2].

On the Sum and Spread of Reciprocal Distance Laplacian Eigenvalues of Graphs in Terms of Harary Index

The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G)=RT(G)−RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Clearly, RDL(G) is a real symmetric matrix, and we denote its eigenvalues as λ1(RDL(G))≥λ2(RDL(G))≥…≥λn(RDL(G)). The largest eigenvalue λ1(RDL(G)) of RDL(G), denoted by λ(G), is called the reciprocal distance Laplacian spectral radius. In this paper, we obtain several upper bounds for the sum of k largest reciprocal distance Laplacian eigenvalues of G in terms of various graph parameters, such as order n, maximum reciprocal distance degree RTmax, minimum reciprocal distance degree RTmin, and Harary index H(G) of G. We determine the extremal cases corresponding to these bounds. As a consequence, we obtain the upper bounds for reciprocal distance Laplacian spectral radius λ(G) in terms of the parameters as mentioned above and characterize the extremal cases. Moreover, we attain several upper and lower bounds for reciprocal distance Laplacian spread RDLS(G)=λ1(RDL(G))−λn−1(RDL(G)) in terms of various graph parameters. We determine the extremal graphs in many cases.

On normalized distance Laplacian eigenvalues of graphs and applications to graphs defined on groups and rings

The normalized distance Laplacian matrix of a connected graph $ G $, denoted by $ D^{\mathcal{L}}(G) $, is defined by $ D^{\mathcal{L}}(G)=Tr(G)^{-1/2}D^L(G)Tr(G)^{-1/2}, $ where $ D(G) $ is the distance matrix, the $D^{L}(G)$ is the distance Laplacian matrix and $ Tr(G)$ is the diagonal matrix of vertex transmissions of $ G. $ The set of all eigenvalues of $ D^{\mathcal{L}}(G) $ including their multiplicities is the normalized distance Laplacian spectrum or $ D^{\mathcal{L}} $-spectrum of $G$. In this paper, we find the $ D^{\mathcal{L}} $-spectrum of the joined union of regular graphs in terms of the adjacency spectrum and the spectrum of an auxiliary matrix. As applications, we determine the $ D^{\mathcal{L}} $-spectrum of the graphs associated with algebraic structures. In particular, we find the $ D^{\mathcal{L}} $-spectrum of the power graphs of groups, the $ D^{\mathcal{L}} $-spectrum of the commuting graphs of non-abelian groups and the $ D^{\mathcal{L}} $-spectrum of the zero-divisor graphs of commutative rings. Several open problems are given for further work.

On the Sum of Distance Laplacian 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 $G$.

Signless Laplacian energy, distance Laplacian energy and distance signless Laplacian spectrum of unitary addition Cayley graphs

In this paper we compute bounds for signless Laplacian energy, distance signless Laplacian eigenvalues and signless Laplacian energy of unitary addition Cayley graph . We also obtain distance Laplacian eigenvalues and distance Laplacian energy of .

On distance Laplacian spectral radius and chromatic number of graphs

Let G be a connected simple graph with n vertices having chromatic number χ. The distance Laplacian matrix DL(G) is defined as DL(G)=Diag(Tr)−D, where Diag(Tr) is the diagonal matrix of vertex transmissions and D is the distance matrix of G. The eigenvalues of DL(G) are the distance Laplacian eigenvalues of G and are denoted by ∂1L(G),∂2L(G),…,∂nL(G). The largest eigenvalue ∂1L(G) is called the distance Laplacian spectral radius. For a non-complete graph G with n vertices and chromatic number χ, Aouchiche and Hansen (2017) proved that ∂1L(G)≥n+⌈nχ⌉. If G is a connected graph with n≥4 vertices and chromatic number χ≤n−2, we prove that ∂2L(G)≥n+⌈nχ⌉ and we show the existence of graphs for which the equality holds. Among all graphs with chromatic number χ satisfying n2≤χ≤n−1, we show that the graph K2,2,…,2︸n−χ,1,1,…,1︸2χ−n has the minimum distance Laplacian spectral radius. Also, we give the distribution of the distance Laplacian eigenvalues in relation to the chromatic number χ and other graph invariants. We characterize the extremal graphs for some of these results and for others, we illustrate by examples.

On the distance Laplacian energy ordering of a tree

For a simple connected graph G of order n having distance Laplacian eigenvalues ρ1L≥ρ2L≥⋯≥ρnL, the distance Laplacian energy DLE(G) is defined as DLE(G)=∑i=1n|ρiL−2W(G)n|, where W(G) is the Weiner index of G. In this paper, we describe the distance Laplacian eigenvalues of a tree of diameter 3. We discuss the distance Laplacian energy of trees of diameter 3 and show that like the Laplacian energy [31] these trees can be ordered on the basis of their distance Laplacian energy. As application we obtain an upper bound for the distance Laplacian energy of a connected graph.

On distance Laplacian spectrum (energy) of graphs

For a simple connected graph [Formula: see text] of order [Formula: see text] having distance Laplacian eigenvalues [Formula: see text], the distance Laplacian energy [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the Wiener index of [Formula: see text]. We obtain the distance Laplacian spectrum of the joined union of graphs [Formula: see text] in terms of their distance Laplacian spectrum and the spectrum of an auxiliary matrix. As application, we obtain the distance Laplacian spectrum of the lexicographic product of graphs. We study the distance Laplacian energy of connected graphs with given chromatic number [Formula: see text]. We show that among all connected graphs with chromatic number [Formula: see text] the complete [Formula: see text]-partite graph has the minimum distance Laplacian energy. Further, we discuss the distribution of distance Laplacian eigenvalues around average transmission degree [Formula: see text].

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

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.

Distance matrices on the H-join of graphs: A general result and applications

Given a graph H with vertices 1,…,s and a set of pairwise vertex disjoint graphs G1,…,Gs, the vertex i of H is assigned to Gi. Let G be the graph obtained from the graphs G1,…,Gs and the edges connecting each vertex of Gi with all the vertices of Gj for all edge ij of H. The graph G is called the H-join of G1,…,Gs. Let M(G) be a matrix on a graph G. A general result on the eigenvalues of M(G), when the all ones vector is an eigenvector of M(Gi) for i=1,2,…,s, is given. This result is applied to obtain the distance eigenvalues, the distance Laplacian eigenvalues and as well as the distance signless Laplacian eigenvalues of G when G1,…,Gs are regular graphs. Finally, we introduce the notions of the distance incidence energy and distance Laplacian-energy like of a graph and we derive sharp lower bounds on these two distance energies among all the connected graphs of prescribed order in terms of the vertex connectivity. The graphs for which those bounds are attained are characterized.

Relation between the matching number and the second largest distance Laplacian eigenvalue of a graph

Let G be a connected simple graph with matching number m(G). The second largest distance Laplacian eigenvalue of G is denoted by ∂2(G). In this paper, we investigate the relation between the matching number and the second largest distance Laplacian eigenvalue of G, establishing the lower bounds of ∂2(G) in terms of m(G). Moreover, all the extremal graphs attaining the lower bounds are completely characterized.

Graphs with some distance Laplacian eigenvalue of multiplicity n − 3

The eigenvalues of the distance Laplacian matrix of a connected graph G are called the distance Laplacian eigenvalues of G. Denote by G(n,n−3) the set of all connected graphs on n vertices with some distance Laplacian eigenvalue of multiplicity n−3. In this paper, all graphs belonging to G(n,n−3) are characterized completely, which solves the remaining problems in Fernandes et al. (2018) [11].

On distance Laplacian and distance signless Laplacian eigenvalues of graphs

ABSTRACTLet , and be, respectively, the distance matrix, the distance Laplacian matrix and the distance signless Laplacian matrix of graph G, where denotes the diagonal matrix of the vertex transmissions in G. The eigenvalues of and will be denoted by and , respectively. In this paper, we study the properties of the distance Laplacian eigenvalues and the distance signless Laplacian eigenvalues of graph G.