154 research outputs found

    Computing the reciprocal distance signless Laplacian eigenvalues and energy of graphs

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    ‎In this paper‎, ‎we study the eigenvalues of the reciprocal distance signless Laplacian matrix of a connected graph and‎ ‎obtain some bounds for the maximum‎ ‎eigenvalue of this matrix‎. ‎We also focus on bipartite graphs and find some bounds for the spectral radius of the reciprocal distance signless Laplacian matrix of this class of graphs‎. ‎Moreover‎, ‎we give bounds for the reciprocal distance signless Laplacian energy

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

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    Given a graph HH with vertices 1,…,s1,\ldots ,s and a set of pairwise vertex disjoint graphs G1,…,Gs,G_{1},\ldots ,G_{s}, the vertex ii of HH is assigned to Gi.G_{i}. Let GG be the graph obtained from the graphs G1,…,GsG_{1},\ldots ,G_{s} and the edges connecting each vertex of GiG_{i} with all the vertices of GjG_{j} for all edge ijij of H.H. The graph GG is called the H−joinH-join of G1,…,GsG_1,\ldots,G_s. Let M(G)M(G) be a matrix on a graph GG. A general result on the eigenvalues of M(G)M\left( G\right) , when the all ones vector is an eigenvector of M(Gi)M\left( G_{i}\right) for i=1,2,…,si=1,2,\ldots ,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 GG when G1,…,GsG_{1},\ldots ,G_{s} 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.publishe

    Sharp Bounds on (Generalized) Distance Energy of Graphs

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    Given a simple connected graph G, let D(G) be the distance matrix, DL(G) be the distance Laplacian matrix, DQ(G) be the distance signless Laplacian matrix, and Tr(G) be the vertex transmission diagonal matrix of G. We introduce the generalized distance matrix Dα(G)=αTr(G)+(1−α)D(G) , where α∈[0,1] . Noting that D0(G)=D(G),2D12(G)=DQ(G),D1(G)=Tr(G) and Dα(G)−Dβ(G)=(α−β)DL(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

    On the Generalized Distance Energy of Graphs

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    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
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