# Bulletin of theKorean Mathematical SocietyBKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Published online May 4, 2022

## Complete characterization of odd factors via the size, spectral radius or distance spectral radius of graphs

Shuchao Li and Shujing Miao

Central China Normal University

### Abstract

Given a graph $G,$ a $\{1,3,\ldots,2n-1\}$-factor of $G$ is a spanning subgraph of $G$, in which each degree of vertices is one of $\{1,3,\ldots,2n-1\}$, where $n$ is a positive integer. In this paper, we first establish a lower bound on the size (resp. the spectral radius) of $G$ to guarantee that $G$ contains a $\{1,3,\ldots,2n-1\}$-factor. Then we determine an upper bound on the distance spectral radius of $G$ to ensure that $G$ has a $\{1,3,\ldots,2n-1\}$-factor. Furthermore, we construct some extremal graphs to show all the bounds obtained in this contribution are best possible.