The maximum number of stars in a graph without linear forest

TL;DR

This paper determines the maximum number of star subgraphs in large graphs that avoid a specific linear forest, extending previous results and exploring extremal graph characterizations.

Contribution

It generalizes known extremal results for star and path graphs to broader classes involving linear forests and characterizes the extremal graphs for these cases.

Findings

Derived exact extremal numbers for large n

Characterized extremal graphs attaining these maximums

Extended previous results on specific graph configurations

Abstract

For two graphs $J$ and $H$, the generalized Tur\'{a}n number, denoted by $ex(n,J,H)$, is the maximum number of copies of $J$ in an $H$-free graph of order $n$. A linear forest $F$ is the disjoint union of paths. In this paper, we determine the number $ex(n,S_r,F)$ when $n$ is large enough and characterize the extremal graphs attaining $ex(n,S_r,F)$, which generalizes the results on $ex(n, S_r, P_k)$, $ex(n,K_2,(k+1) P_2)$ and $ex(n,K^*_{1,r},(k+1) P_2)$. Finally, we pose the problem whether the extremal graph for $ex(n,J,F)$ is isomorphic to that for $ex(n,S_r,F)$, where $J$ is any graph such that the number of $J$'s in any graph $G$ does not decrease by shifting operation on $G$.