# Peripheral Hosoya polynomial of composite graphs

Document Type : Research Paper

Authors

Department of Mathematics, Mangalore University, Mangalore-574199, India

Abstract

Peripheral Hosoya polynomial of a graph $G$ is defined as‎,
‎ \begin{align*}‎
‎ &PH(G,\lambda)=\sum_{k\geq 1}d_P(G,k)\lambda^k,\\‎
‎ \text{where $d_P(G,k)$ is the number} &\text{ of pairs of peripheral vertices at distance $k$ in $G$.}‎
‎ \end{align*}‎
Peripheral Hosoya polynomial of composite graphs viz.‎, ‎$G_1\times G_2$ the Cartesian product‎, ‎$G_1+G_2$ the join‎, ‎$G_1[G_2]$ the composition‎, ‎$G_1\circ G_2$ the corona and $G_1\{G_2\}$ the cluster of arbitrary connected graphs $G_1$ and $G_2$ are computed and their peripheral Wiener indices are stated as immediate consequences‎.

Keywords

Main Subjects

#### References

[1] W. Imrich and S. Klavzar, Product graphs: structure and recognition, With a foreword by Peter Winkler. Wiley-
Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York, 2000.
[2] F. Harary, Graph Theory, 2nd printing, (Addison-Wesley, Reading, MA,1971).
[3] R. Frucht and F. Harary, On the Corona of Two Graphs, Aequationes Math., 4 (1970) 322–325.
[4] F. Buckley and F. Harary, Distance in graphs, Addison-Wesley Publishing Company, Advanced Book Program,
Redwood City, CA, 1990.
[5] H. Hosoya, On some counting polynomials in chemistry. Discrete Appl. Math., 19 (1988) 239–257.
[6] K. P. Narayankar, S. B. Lokesh, S. S. Shirkol and H. S. Ramane, Terminal Hosoya polynomial of thorn graphs,
Scientia Magna.
[7] I. Gutman, B. Furtula and M. Petrović, Terminal Wiener index, J. Math. Chem., 46 (2009) 522–531.
[8] Keerthi G. Mirajkar and B. Pooja, On the Hosoya polynomial and Wiener index of jump graph, Jordan J. Math.
Stat., 13 (2020) 37–59.
[9] G. Jayalalitha, M. Raji and S. Senthil, Hosoya polynomial and wiener index of molecular graph of naphthalene
based on domination, International Journal Of Scientific Research And Review, 8 (2019) 126-129.
[10] K. P. Narayankar, S. Lokesh, V. Mathad and I. Gutman, Hosoya polynomial of Hanoi graphs, Kragujevac J. Math.,
36 (2012) 51–57.
[11] D. Shubhalakshimi, Distance parametrs in graphs and its applications, Mangalore University, Mangalore, India,
2017.
[12] K. P. Narayankar and S. B. Lokesh, Peripheral wiener index of a graph. Commun. Comb. Optim., 2 (2017) 43–56.
[13] H. Hua, On the peripheral Wiener index of graphs, Discrete Appl. Math., 258 (2019) 135–142.
[14] Y-H Chen, H. Wang and X-D Zhang, Peripheral wiener index of trees and related questions, Discrete Appl. Math.,
251 (2018) 135–145.
[15] A. Kahsay and K. P. Narayankar, Peripheral wiener index of graph operations, Bull. Int. Math. Virtual Inst., 9
(2019) 591–597.
[16] K. P. Narayankar, A. Kahsay and S. Klavzar, On peripheral wiener index: line graphs, zagreb index, and cut
method, MATCH Commun. Math. Comput. Chem., 83 (2020) 129–141.
[17] K. P. Narayankar and A. Ali, Peripheral Hosoya polynomial of thorn graphs, Indian J. Discrete Math., 6 (2020)
1–13.
[18] K. P. Narayankar, S. B. Lokesh, D. Shubhalakshimi and H. S. Ramane, Peripheral path index polynomial, Indian
J. Discrete Math., 1 (2015) 45–57.
[19] A. J. Schwenk, Computing the characteristic polynomial of a graph, Graphs and combinatorics (Proc. Capital Conf.,
George Washington) Univ., D. C. Washington, Lecture Notes in Math., 406, Springer, Berlin, 1973 153–172.
[20] Y. Yeh and I. Gutman, On the sum of all distances in composite graphs, Discrete Math., 135 (1994) 359–365.
[21] D. Stevanović, Hosoya polynomial of composite graphs, Discrete Math., 235 (2001) 237–244.