| Summary: | For a simple undirected connected graph <i>G</i> of order <i>n</i>, let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>D</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>D</mi> <mi>L</mi> </msup> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>D</mi> <mi>Q</mi> </msup> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>T</mi> <mi>r</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> be, respectively, the distance matrix, the distance Laplacian matrix, the distance signless Laplacian matrix and the diagonal matrix of the vertex transmissions of <i>G</i>. The generalized distance matrix <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>D</mi> <mi>α</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is signified by <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>D</mi> <mi>α</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>α</mi> <mi>T</mi> <mi>r</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo>)</mo> </mrow> <mi>D</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> Here, we propose a new kind of Estrada index based on the Gaussianization of the generalized distance matrix of a graph. Let <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mo>∂</mo> <mn>1</mn> </msub> <mo>,</mo> <msub> <mo>∂</mo> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mo>∂</mo> <mi>n</mi> </msub> </mrow> </semantics> </math> </inline-formula> be the generalized distance eigenvalues of a graph <i>G</i>. We define the generalized distance Gaussian Estrada index <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>P</mi> <mi>α</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, as <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>P</mi> <mi>α</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msup> <mi>e</mi> <mrow> <mo>−</mo> <msubsup> <mo>∂</mo> <mi>i</mi> <mn>2</mn> </msubsup> </mrow> </msup> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> Since characterization of <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>P</mi> <mi>α</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is very appealing in quantum information theory, it is interesting to study the quantity <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>P</mi> <mi>α</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> and explore some properties like the bounds, the dependence on the graph topology <i>G</i> and the dependence on the parameter <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>. In this paper, we establish some bounds for the generalized distance Gaussian Estrada index <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>P</mi> <mi>α</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> of a connected graph <i>G</i>, involving the different graph parameters, including the order <i>n</i>, the Wiener index <inline-formula> <math display="inline"> <semantics> <mrow> <mi>W</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, the transmission degrees and the parameter <inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics> </math> </inline-formula>, and characterize the extremal graphs attaining these bounds.
|
|---|