TY - JOUR

T1 - On Generalized Distance Gaussian Estrada Index of Graphs

AU - Alhevaz, Abdollah

AU - Baghipur, Maryam

AU - Shang, Yilun

PY - 2019/10/11

Y1 - 2019/10/11

N2 - For a simple undirected connected graph G of order n, let D(G) , DL(G) , DQ(G) and Tr(G) be, respectively, the distance matrix, the distance Laplacian matrix, the distance signless Laplacian matrix and the diagonal matrix of the vertex transmissions of G. The generalized distance matrix Dα(G) is signified by Dα(G)=αTr(G)+(1−α)D(G) , where α∈[0,1]. Here, we propose a new kind of Estrada index based on the Gaussianization of the generalized distance matrix of a graph. Let ∂1,∂2,…,∂n be the generalized distance eigenvalues of a graph G. We define the generalized distance Gaussian Estrada index Pα(G) , as Pα(G)=∑ni=1e−∂2i. Since characterization of Pα(G) is very appealing in quantum information theory, it is interesting to study the quantity Pα(G) and explore some properties like the bounds, the dependence on the graph topology G and the dependence on the parameter α . In this paper, we establish some bounds for the generalized distance Gaussian Estrada index Pα(G) of a connected graph G, involving the different graph parameters, including the order n, the Wiener index W(G) , the transmission degrees and the parameter α∈[0,1] , and characterize the extremal graphs attaining these bounds.

AB - For a simple undirected connected graph G of order n, let D(G) , DL(G) , DQ(G) and Tr(G) be, respectively, the distance matrix, the distance Laplacian matrix, the distance signless Laplacian matrix and the diagonal matrix of the vertex transmissions of G. The generalized distance matrix Dα(G) is signified by Dα(G)=αTr(G)+(1−α)D(G) , where α∈[0,1]. Here, we propose a new kind of Estrada index based on the Gaussianization of the generalized distance matrix of a graph. Let ∂1,∂2,…,∂n be the generalized distance eigenvalues of a graph G. We define the generalized distance Gaussian Estrada index Pα(G) , as Pα(G)=∑ni=1e−∂2i. Since characterization of Pα(G) is very appealing in quantum information theory, it is interesting to study the quantity Pα(G) and explore some properties like the bounds, the dependence on the graph topology G and the dependence on the parameter α . In this paper, we establish some bounds for the generalized distance Gaussian Estrada index Pα(G) of a connected graph G, involving the different graph parameters, including the order n, the Wiener index W(G) , the transmission degrees and the parameter α∈[0,1] , and characterize the extremal graphs attaining these bounds.

KW - Gaussian Estrada index

KW - generalized distance matrix (spectrum)

KW - Wiener index

KW - generalized distance Gaussian Estrada index

KW - transmission regular graph

U2 - 10.3390/sym11101276

DO - 10.3390/sym11101276

M3 - Article

SN - 2073-8994

VL - 11

JO - Symmetry

JF - Symmetry

IS - 10

M1 - 1276

ER -