Unicyclic Graphs and Their Zagreb Indices: A Subdivision Approach
Unicyclic Graphs and Their Zagreb Indices: A Subdivision Approach
Abstract
The first Zagreb index M1(G) of a graph G is defined as the sum of the squares of the degrees of its vertices, while the second Zagreb index M2(G) is given by the sum of the products of degrees of all adjacent vertex pairs. In this work, we investigate these indices and their corresponding coindices for a specific family of unicyclic graphs, referred to as cycle-star graphs. Our approach employs the concept of graph subdivision to analyze the Zagreb indices for both the line graph and the line cut-vertex graph derived from this graph class.
Downloads
Download data is not yet available.
References
\bibitem{1}{T. Do\v{s}li\'c, \emph{Vertex-weighted Weiner polynomials for composite graphs}, Ars Math. Contemp. \textbf{1}, 66-80, 2008}.
\bibitem{2}{K. C. Das and I. Gutman, \emph{Some properties of the second Zagreb index}, MATCH Commun. Math. Comput. Chem. \textbf{52}, 103-112, 2004}.
\bibitem{3}{B. Furtula, I. Gutman and M. Dehmer, \emph{On structure-sensitivity of degree-based topological indices}, Appl. Math. Comput. \textbf{219}, 8973-8978, 2013}.
\bibitem{4}{I. Gutman and N. Trinajsti\'c N, \emph{Graph theory and molecular orbitals. Total $\pi$-electron energy of alternant hydrocarbons}, Chem Phys Lett. \textbf{17}, 535-538, 1972}.
\bibitem{5}{I. Gutman, B. Ru\v{s}\v{c}i\'c, N. Trinajsti\'c and C. F. Wilcox, \emph{Graph theory and molecular orbitals. XII. Acyclic polyenes}, J Chem Phys. \textbf{62}, 3399-3405, 1975}.
\bibitem{6}{I. Gutman, \emph{Degree-based topological indices}, Croat. Chem, Acta. \textbf{86}, 351-361, 2013}.
\bibitem{7}{I. Gutman and K. C. Das, \emph{The first Zagreb index 30 years after}, MATCH Commun. Math. Comput. Chem. \textbf{50}, 83-92, 2004}.
\bibitem{8}{I. Gutman, B. Furtula, Z. K. Vuki\'cevi\'c and G. Popivoda, \emph{On Zagreb Indices and Coindices}, MATCH Commun. Math. Comput. Chem. \textbf{74}, 5-16, 2015}.
\bibitem{9}{I. Gutman and J.To\v{s}vi\'c, \emph{Testing the quality of molecular structure descriptors: Vertex-degree based topological indices}, J. Serb. Chem. Soc. \textbf{78}, 805-810, 2013}.
\bibitem{10}{I. Gutman, \emph{On the origin of two degree-based topological indices}, Bull. Acad. Serbe Sci. Arts (Cl. Sci. Math. Natur.). \textbf{146}, 39-52, 2014}.
\bibitem{11}{F. Harary, \emph{Graph Theory}, Addison-Wesley, Reading, Mass (1969)}.
\bibitem{12}{V. R. Kulli and M. H. Muddebihal, \emph{On lict and litact graph of a graph}, Proceeding of the Indian National Science Academy. \textbf{41}, 275-280, 1975}.
\bibitem{13}{M. R. Alfuraidan, Tomas Vetr\'i k and S. Balachandran, \emph{General multiplicative Zagreb indices of unicyclic graphs}, Carpathian Journal of Mathematics. \textbf{37}, 1-13, 2021}.
\bibitem{14}{S. Nikoli\'c, G. Kova\v{c}evi\'c, A. Mili\'cevi\'c and N. Trinajsti\'c, \emph{The Zagreb indices 30 years after}, Croat Chem Acta. \textbf{76}, 113-124, 2003}.
\bibitem{15}{H. M. Nagesh, \emph{On the Zagreb indices of the line cut-vertex graph of subdivision of graphs}, Bull. Int. Math. Virtual Inst. \textbf{12}, 17-26, 2022}.
\bibitem{16}{H. M. Nagesh and V. R. Girish, \emph{On the entire Zagreb indices of the line graph and line cut-vertex graph of the subdivision graph}, Open Journal of Mathematical Sciences. \textbf{4(2)}, 470-475, 2020.}
\bibitem{17}{H. M. Nagesh, V. R. Girish and B. Azghar Pasha, \emph{On the entire Zagreb indices of the line graph and line cut-vertex graph of the subdivision graph}, Open Journal of Mathematical Sciences. \textbf{12(1)}, 159-167, 2022}.
\bibitem{18}{E. Prisner, \emph{Graph dynamics}, Chapman \& Hall/CRC, \textbf{338} (1995)}.
\bibitem{19}{P. S. Ranjini, V. Lokesha and I. N. Kangul, \emph{On the Zabreb indices of the line graphs of the subdivision graphs}, Appl. Math. Comput. \textbf{210}, 699-702, 2011}.
\bibitem{20}{J. Sedlar, \emph{Extremal unicyclic graphs with respect to additively weighted Harary index}, Miskolic mathematical Notes. \textbf{16(2)}, 1-16, 2013}.
\bibitem{21}{Xuli Qi, Bo Zhou and Jiyong Li, \emph{Zagreb eccentricity indices of unicyclic graphs}, Discrete Applied Mathematics. \textbf{233(2)}, 166-174, 2017}.
\bibitem{22}{Zikai Tang and Hechao Liu, \emph{Maximal hyper-Zagreb index of trees, unicyclic and bicyclic graphs with a given order and
matching number}, Discrete Math. Lett. \textbf{4}, 11-18, 2020}.
\bibitem{2}{K. C. Das and I. Gutman, \emph{Some properties of the second Zagreb index}, MATCH Commun. Math. Comput. Chem. \textbf{52}, 103-112, 2004}.
\bibitem{3}{B. Furtula, I. Gutman and M. Dehmer, \emph{On structure-sensitivity of degree-based topological indices}, Appl. Math. Comput. \textbf{219}, 8973-8978, 2013}.
\bibitem{4}{I. Gutman and N. Trinajsti\'c N, \emph{Graph theory and molecular orbitals. Total $\pi$-electron energy of alternant hydrocarbons}, Chem Phys Lett. \textbf{17}, 535-538, 1972}.
\bibitem{5}{I. Gutman, B. Ru\v{s}\v{c}i\'c, N. Trinajsti\'c and C. F. Wilcox, \emph{Graph theory and molecular orbitals. XII. Acyclic polyenes}, J Chem Phys. \textbf{62}, 3399-3405, 1975}.
\bibitem{6}{I. Gutman, \emph{Degree-based topological indices}, Croat. Chem, Acta. \textbf{86}, 351-361, 2013}.
\bibitem{7}{I. Gutman and K. C. Das, \emph{The first Zagreb index 30 years after}, MATCH Commun. Math. Comput. Chem. \textbf{50}, 83-92, 2004}.
\bibitem{8}{I. Gutman, B. Furtula, Z. K. Vuki\'cevi\'c and G. Popivoda, \emph{On Zagreb Indices and Coindices}, MATCH Commun. Math. Comput. Chem. \textbf{74}, 5-16, 2015}.
\bibitem{9}{I. Gutman and J.To\v{s}vi\'c, \emph{Testing the quality of molecular structure descriptors: Vertex-degree based topological indices}, J. Serb. Chem. Soc. \textbf{78}, 805-810, 2013}.
\bibitem{10}{I. Gutman, \emph{On the origin of two degree-based topological indices}, Bull. Acad. Serbe Sci. Arts (Cl. Sci. Math. Natur.). \textbf{146}, 39-52, 2014}.
\bibitem{11}{F. Harary, \emph{Graph Theory}, Addison-Wesley, Reading, Mass (1969)}.
\bibitem{12}{V. R. Kulli and M. H. Muddebihal, \emph{On lict and litact graph of a graph}, Proceeding of the Indian National Science Academy. \textbf{41}, 275-280, 1975}.
\bibitem{13}{M. R. Alfuraidan, Tomas Vetr\'i k and S. Balachandran, \emph{General multiplicative Zagreb indices of unicyclic graphs}, Carpathian Journal of Mathematics. \textbf{37}, 1-13, 2021}.
\bibitem{14}{S. Nikoli\'c, G. Kova\v{c}evi\'c, A. Mili\'cevi\'c and N. Trinajsti\'c, \emph{The Zagreb indices 30 years after}, Croat Chem Acta. \textbf{76}, 113-124, 2003}.
\bibitem{15}{H. M. Nagesh, \emph{On the Zagreb indices of the line cut-vertex graph of subdivision of graphs}, Bull. Int. Math. Virtual Inst. \textbf{12}, 17-26, 2022}.
\bibitem{16}{H. M. Nagesh and V. R. Girish, \emph{On the entire Zagreb indices of the line graph and line cut-vertex graph of the subdivision graph}, Open Journal of Mathematical Sciences. \textbf{4(2)}, 470-475, 2020.}
\bibitem{17}{H. M. Nagesh, V. R. Girish and B. Azghar Pasha, \emph{On the entire Zagreb indices of the line graph and line cut-vertex graph of the subdivision graph}, Open Journal of Mathematical Sciences. \textbf{12(1)}, 159-167, 2022}.
\bibitem{18}{E. Prisner, \emph{Graph dynamics}, Chapman \& Hall/CRC, \textbf{338} (1995)}.
\bibitem{19}{P. S. Ranjini, V. Lokesha and I. N. Kangul, \emph{On the Zabreb indices of the line graphs of the subdivision graphs}, Appl. Math. Comput. \textbf{210}, 699-702, 2011}.
\bibitem{20}{J. Sedlar, \emph{Extremal unicyclic graphs with respect to additively weighted Harary index}, Miskolic mathematical Notes. \textbf{16(2)}, 1-16, 2013}.
\bibitem{21}{Xuli Qi, Bo Zhou and Jiyong Li, \emph{Zagreb eccentricity indices of unicyclic graphs}, Discrete Applied Mathematics. \textbf{233(2)}, 166-174, 2017}.
\bibitem{22}{Zikai Tang and Hechao Liu, \emph{Maximal hyper-Zagreb index of trees, unicyclic and bicyclic graphs with a given order and
matching number}, Discrete Math. Lett. \textbf{4}, 11-18, 2020}.
Published
2026-04-01
Section
Research Articles
Copyright (c) 2026 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
