AUTHOR=Yasmeen Farhana , Akhter Shehnaz , Ali Kashif , Rizvi Syed Tahir Raza 

TITLE=Edge Mostar Indices of Cacti Graph With Fixed Cycles

JOURNAL=Frontiers in Chemistry

VOLUME=9

YEAR=2021

URL=https://www.frontiersin.org/journals/chemistry/articles/10.3389/fchem.2021.693885

DOI=10.3389/fchem.2021.693885

ISSN=2296-2646

ABSTRACT=<p>Topological invariants are the significant invariants that are used to study the physicochemical and thermodynamic characteristics of chemical compounds. Recently, a new bond additive invariant named the Mostar invariant has been introduced. For any connected graph <inline-formula id="inf1"><mml:math id="m1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">ℋ</mml:mi></mml:math></inline-formula>, the edge Mostar invariant is described as <inline-formula id="inf2"><mml:math id="m2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>M</mml:mi><mml:msub><mml:mi>o</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi mathvariant="normal">ℋ</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>g</mml:mi><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi mathvariant="normal">ℋ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:munder><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mi mathvariant="normal">ℋ</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>g</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mi mathvariant="normal">ℋ</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:math></inline-formula>, where <inline-formula id="inf3"><mml:math id="m3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mi mathvariant="normal">ℋ</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>g</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext>or </mml:mtext><mml:msub><mml:mi>m</mml:mi><mml:mi mathvariant="normal">ℋ</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> is the number of edges of <inline-formula id="inf4"><mml:math id="m4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">ℋ</mml:mi></mml:math></inline-formula> lying closer to vertex <italic>g</italic> (or <italic>x</italic>) than to vertex <italic>x</italic> (or <italic>g</italic>). A graph having at most one common vertex between any two cycles is called a cactus graph. In this study, we compute the greatest edge Mostar invariant for cacti graphs with a fixed number of cycles and <italic>n</italic> vertices. Moreover, we calculate the sharp upper bound of the edge Mostar invariant for cacti graphs in <inline-formula id="inf5"><mml:math id="m5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">ℭ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>, where <italic>s</italic> is the number of cycles.</p>