On Generalized Distance Spectral Radius of a Bipartite Graph (original) (raw)
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On spectral spread of generalized distance matrix of a graph
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For a simple connected graph G, let D(G), T r(G), D L (G) and D Q (G), respectively be the distance matrix, the diagonal matrix of the vertex transmissions, distance Laplacian matrix and the distance signless Laplacian matrix of a graph G. The convex linear combinations D α (G) of T r(G) and D(G) is defined as D α (G) = αT r(G) + (1 − α)D(G), 0 ≤ α ≤ 1. As D 0 (G) = D(G), 2D 1 2 (G) = D Q (G), D 1 (G) = T r(G) and D α (G) − D β (G) = (α − β)D L (G), this matrix reduces to merging the distance spectral, distance Laplacian spectral and distance signless Laplacian spectral theories. Let ∂ 1 (G) ≥ ∂ 2 (G) ≥ • • • ≥ ∂ n (G) be the eigenvalues of D α (G) and let D α S(G) = ∂ 1 (G) −∂ n (G) be the generalized distance spectral spread of the graph G. In this paper, we obtain some bounds for the generalized distance spectral spread D α (G). We also obtain relation between the generalized distance spectral spread D α (G) and the distance spectral spread S D (G). Further, we obtain the lower bounds for D α S(G) of bipartite graphs involving different graph parameters and we characterize the extremal graphs for some cases. We also obtain lower bounds for D α S(G) in terms of clique number and independence number of the graph G and characterize the extremal graphs for some cases.
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Bounds for the Generalized Distance Eigenvalues of a Graph
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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 s...
Bounds for Generalized Distance Spectral Radius and the Entries of the Principal Eigenvector
Tamkang Journal of Mathematics
For a simple connected graph GGG, the convex linear combinations Dalpha(G)D_{\alpha}(G)Dalpha(G) of \ Tr(G)Tr(G)Tr(G) and D(G)D(G)D(G) is defined as Dalpha(G)=alphaTr(G)+(1−alpha)D(G)D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G)Dalpha(G)=alphaTr(G)+(1−alpha)D(G), 0leqalphaleq10\leq \alpha\leq 10leqalphaleq1. As D0(G)=D(G)D_{0}(G)=D(G)D0(G)=D(G), 2Dfrac12(G)=DQ(G)2D_{\frac{1}{2}}(G)=D^{Q}(G)2Dfrac12(G)=DQ(G), D1(G)=Tr(G)D_{1}(G)=Tr(G)D1(G)=Tr(G) and Dalpha(G)−Dbeta(G)=(alpha−beta)DL(G)D_{\alpha}(G)-D_{\beta}(G)=(\alpha-\beta)D^{L}(G)Dalpha(G)−Dbeta(G)=(alpha−beta)DL(G), this matrix reduces to merging the distance spectral and distance signless Laplacian spectral theories. In this paper, we study the spectral properties of the generalized distance matrix Dalpha(G)D_{\alpha}(G)Dalpha(G). We obtain some lower and upper bounds for the generalized distance spectral radius, involving different graph parameters and characterize the extremal graphs. Further, we obtain upper and lower bounds for the maximal and minimal entries of the $ p −normnormalizedPerronvectorcorrespondingtospectralradius-norm normalized Perron vector corresponding to spectral radius −normnormalizedPerronvectorcorrespondingtospectralradius \partial(G) $ of the generalized distance matrix Dalpha(G)D_{\alpha}(G)Dalpha(G) and characterize the extremal graphs.
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The Laplacian spectral radius of a graph is the largest eigenvalue of the associated Laplacian matrix. In this paper, we determine those graphs which maximize the Laplacian spectral radius among all bipartite graphs with (edge-)connectivity at most k. We also characterize graphs of order n with k cut-edges, having Laplacian spectral radius equal to n.
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The Laplacian spectral radius of a graph is the largest eigenvalue of the associated Laplacian matrix. In this paper, we determine those graphs which maximize the Laplacian spectral radius among all bipartite graphs with (edge-)connectivity at most k. We also characterize graphs of order n with k cut-edges, having Laplacian spectral radius equal to n.
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Let G be a simple graph with n vertices and m edges and G be its complement. Let δ(G) = δ and ∆(G) = ∆ be the minimum degree and the maximum degree of vertices of G, respectively. In this paper we present a sharp upper bound for the Laplacian spectral radius of complete bipartite graph Kr,s for s ≥ r as λ1(G) ≤ s+ √ s2+8rs 2 and for r = s, λ1(G) = 2s. We prove the upper bound of Nordhaus Gaddum type relation on the Laplacian spectral radius of this graph as λ1(G) + λ1(G ) ≤ s+ √ s2+8rs 2 + (s− 1) ≤ (3s− 1).
On the spread of the distance signless Laplacian matrix of a graph
Acta Universitatis Sapientiae, Informatica
Let G be a connected graph with n vertices, m edges. The distance signless Laplacian matrix DQ(G) is defined as DQ(G) = Diag(Tr(G)) + D(G), where Diag(Tr(G)) is the diagonal matrix of vertex transmissions and D(G) is the distance matrix of G. The distance signless Laplacian eigenvalues of G are the eigenvalues of DQ(G) and are denoted by δ1 Q(G), δ2 Q(G), ..., δn Q(G). δ1 Q is called the distance signless Laplacian spectral radius of DQ(G). In this paper, we obtain upper and lower bounds for SDQ (G) in terms of the Wiener index, the transmission degree and the order of the graph.
On Complementary Distance Signless Laplacian Spectral Radius and Energy of Graphs
Iranian Journal of Mathematical Sciences and Informatics, 2019
Harishchandra S. Ramane, Gouramma A. Gudodagi, Vinayak V. Manjalapurc,∗ and Abdollah Alhevaz Department of Mathematics, Karnatak University, Dahrwad580003, India. Department of Mathematics, KLE Societys, G. I. Bagewadi Arts, Science and Commerce College, Nipani 591237, Karnataka, India. Department of Mathematics, KLE Societys, Basavaprabhu Kore Arts, Science and Commerce College, Chikodi 591201, Karnataka, India. Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box: 316-3619995161, Shahrood, Iran.
Sharp Bounds on (Generalized) Distance Energy of Graphs
Mathematics, 2020
Given a simple connected graph G, let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian matrix, D Q ( G ) be the distance signless Laplacian matrix, and T r ( G ) be the vertex transmission diagonal matrix of G. We introduce the generalized distance matrix D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where α ∈ [ 0 , 1 ] . Noting that D 0 ( G ) = D ( G ) , 2 D 1 2 ( G ) = D Q ( G ) , D 1 ( G ) = T r ( G ) and D α ( G ) − D β ( G ) = ( α − β ) D L ( G ) , we reveal that a generalized distance matrix ideally bridges the spectral theories of the three constituent matrices. In this paper, we obtain some sharp upper and lower bounds for the generalized distance energy of a graph G involving different graph invariants. As an application of our results, we will be able to improve some of the recently given bounds in the literature for distance energy and distance signless Laplacian energy of graphs. The extremal graphs of the corresponding bounds are also characterized.