AUTHOR=Fang Wei , Ma Muhan , Chen FuYuan , Dong Hufeng 

TITLE=Third Smallest Wiener Polarity Index of Unicyclic Graphs

JOURNAL=Frontiers in Physics

VOLUME=8

YEAR=2020

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2020.553261

DOI=10.3389/fphy.2020.553261

ISSN=2296-424X

ABSTRACT=<p>The Wiener polarity index <inline-formula id="inf1"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>W</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> of a graph <italic>G</italic> is the number of unordered pairs of vertices <inline-formula id="inf2"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> where the distance between <italic>u</italic> and <italic>v</italic> is 3. In this paper, we determine the third smallest Wiener polarity index of unicyclic graphs. Moreover, the corresponding extremal graphs are characterized.</p>