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.

\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**

- Peripheral Hosoya polynomial
- Composite graph
- Peripheral Wiener index
- Hosoya polynomial
- Wiener Index

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June 2022

Pages 63-76

**Receive Date:**23 January 2021**Revise Date:**05 October 2021**Accept Date:**12 October 2021**Published Online:**01 June 2022