# Spectral extrema of graphs: Forbidden star-path forests

## TL;DR

This paper determines the maximum spectral radius of large graphs that exclude specific star-path forest subgraphs, advancing spectral graph theory by identifying extremal structures under these constraints.

## Contribution

It introduces new extremal spectral bounds for graphs forbidding certain star-path forest configurations, extending previous results in spectral extremal graph theory.

## Key findings

- Identifies maximum spectral radius for graphs avoiding $kS_{\ell-1}\cup P_{\ell}$
- Determines spectral bounds for graphs excluding $k_1S_{2\ell -1}\cup k_2P_{2\ell}$
- Establishes extremal graphs for forbidden star-path forests

## Abstract

A path of order $n$ is denoted by $P_n$, and a star of order $n$ is denoted by $S_{n-1}$. A star-path forest is a forest whose connected components are paths and stars. In this paper we determine the maximum spectral radius of graphs that contain no copy of $kS_{\ell-1}\cup P_{\ell}$, $k_1S_{2\ell -1}\cup k_2P_{2\ell}$ or $kS_{4}\cup 2P_{5}$ for $n$ appropriately large.