Type: Article
Publication Date: 2020-06-11
Citations: 8
DOI: https://doi.org/10.1186/s13660-020-02427-4
Abstract For a connected graph G and $\alpha \in [0,1)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi><mml:mo>∈</mml:mo><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:math> , the distance α -spectral radius of G is the spectral radius of the matrix $D_{\alpha }(G)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>D</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:math> defined as $D_{\alpha }(G)=\alpha T(G)+(1-\alpha )D(G)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>D</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>α</mml:mi><mml:mi>T</mml:mi><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo><mml:mi>D</mml:mi><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:math> , where $T(G)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:math> is a diagonal matrix of vertex transmissions of G and $D(G)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:math> is the distance matrix of G . We give bounds for the distance α -spectral radius, especially for graphs that are not transmission regular, propose local graft transformations that decrease or increase the distance α -spectral radius, and determine the graphs that minimize and maximize the distance α -spectral radius among several families of graphs.