The vertex PI index $PI(G) = \sum_{xy \in E(G)} [n_{xy}(x) + n_{xy}(y)]$ is a distance-based molecular structure descriptor, where $n_{xy}(x)$ denotes the number of vertices which are closer to the vertex $x$ than to the vertex $y$ and which has been the considerable research in computational chemistry dating back to Harold Wiener in 1947. A connected graph is a cactus if any two of its cycles have at most one common vertex. In this paper, we completely determine the extremal graphs with the greatest and smallest vertex PI indices mong all cacti with a fixed number of vertices. As a consequence, we obtain the sharp bounds with corresponding extremal cacti and extend a known result.

[1] T. Al-Fozan, P. Manuel, I. Rajasingh and R. S. Rajan, Computing Szeged index of certain nanosheets using partition technique, MATCH Commun. Math. Comput. Chem., 72 (2014) 339–353.

[2] A. R. Ashrafi and A. Loghman, PI index of zig-zag polyhex nanotubes, MATCH Commun. Math. Comput. Chem., 55 (2006) 447–452.

[3] A. R. Ashrafi and A. Loghman, Padmakar-Ivan index of TUC4C8(S) nanotubes, J. Comput. Theor. Nanosci., 3 (2006) 378–381.

[4] A. R. Ashrafi and A. Loghman, PI index of armchair polyhex nanotubes, Ars Combin., 80 (2006) 193–199.

[5] A. R. Ashrafi, B. Manoochehrian and H. Yousefi-Azari, On the PI polynomial of a graph, Util. Math., 71 (2006) 97–108.

[6] A. R. Ashrafi and F. Rezaei, PI index of polyhex nanotori, MATCH Commun. Math. Comput. Chem., 57 (2007) 243–250.

[7] S. Chen, Cacti with the smallest, second smallest and third smallest Gutman index, J. Comb. Optim., 31 (2016) 327–332.

[8] A. A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math., 66 (2001) 211–249.

[9] A. A. Dobrynin, I. Gutman, S. Klavzzar and P. zZigert, Wiener index of hexagonal systems, Acta Appl. Math., 72 (2002) 247–294.

[10] K. C. Das and I. Gutman, Bound for vertex PI index in terms of simple graph parameters, Filomat, 27 (2013) 1583–1587.

[11] L. Feng and G. Yu, On the hyper-Wiener index of cacti, Util. Math., 93 (2014) 57–64.

[12] I. Gutman, S. Klavzar and B. Mohar, Fiftieth Anniversary of the Wiener Index, Discrete Appl. Math., 80 (1997) 1–113.

[13] M. Hoji, Z. Luo and E. Vumar, Wiener and vertex PI indices of Kronecker products of graphs, Discrete Appl. Math., 158 (2010) 1848–1855.

[14] A. Ilic and N. Milosavljevic, The weighted vertex PI index, Mathematical and Computer Modelling., 57 (2013) 623–631.

[15] S. Klavzar and I. Gutman, The Szeged and the Wiener Index of Graphs, Appl. Math. Lett., 9 (1996) 45–49.

[16] P. V. Khadikar, On a Novel Structural Descriptor PI, Nat. Acad. Sci. Lett., 23 (2000) 113–118.

[17] P. V. Khadikar, P. P. Kale, N. V. Deshpande, S. Karmarkar and V. K. Agrawal, Novel PI indices of hexagonal chains, J. Math. Chem., 29 (2001) 143–150.

[18] P. V. Khadikar, S. Karmarkar and R. G. Varma, The estimation of PI index of polyacenes, Acta Chim. Slov., 49 (2002) 755–771.

[19] M. H. Khalifeh, H. Yousefi-Azari and A. R. Ashrafi, Vertex and edge PI indices of Cartesian product graphs, Discrete Appl. Math., 156 (2008) 1780–1789.

[20] S. Li, H. Yang and Q. Zhao, Sharp bounds on Zagreb indices of cacti with $k$ pendant vertices, Filomat, 26 (2012) 1189–1200.

[21] K. Pattabiraman and P. Paulraja, Wiener and vertex PI indices of the strong product of graphs, Discuss. Math. Graph Theory, 32 (2012) 749–769.

[22] D. Wang and S. Tan, The maximum hyper-Wiener index of cacti, J. Appl. Math. Comput., 47 (2015) 91–102.

[23] H. Wang and L. Kang, On the Harary index of cacti, Util. Math., 96 (2015) 149–163.

[24] H. Wiener, Structural Determination of Paraffin Boiling Points, J. Am. Chem. Soc., 69 (1947) 17–20.

[25] S. Wang and B. Wei, Multiplicative Zagreb indices of cacti, Discrete Math. Algorithm. Appl., DOI:10.1142/S1793830916500403.

[26] S. Wang and B. Wei, Multiplicative Zagreb indices of $k$-trees, Discrete Appl. Math., 180 (2015) 168–175.