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Matchings in graphs from the spectral radius
Linear and Multilinear Algebra ( IF 0.9 ) Pub Date : 2022-05-18 , DOI: 10.1080/03081087.2022.2076799
Minjae Kim 1 , Suil O 2 , Wooyong Sim 1 , Dongwoo Shin 1
Affiliation  

The matching number of G, written α(G), is the size of a maximum matching in G. Suppose that n and k are positive integers of the same parity. Let θ(n,k) be the largest root of x3(nk2)x2(n1)x+k(nk2)=0 and θ(n,k)={θ(n,k)ifn3k+2nk2+(nk2)2+4(n2k2)4ifn3k.In this article, we prove that for a positive integer nk+2, if G is an n-vertex connected graph with the spectral radius ρ(G)>θ(n,k), then α(G)>nk2. The bound is sharp in the sense that for every positive integer nk+2, there are graphs H with ρ(H)=θ(n,k) and α(G)=nk2.



中文翻译:

光谱半径图中的匹配

G的匹配数,写为αG,是G中最大匹配的大小。假设nk是相同奇偶校验的正整数。让θ ( n , k )为最大根X3-n-k-2X2-n-1X+kn-k-2=0θn,k={θn,k如果n3k+2n-k-2+n-k-22+4n2-k24如果n3k在本文中,我们证明对于正整数nk+2,如果G是具有谱半径的n顶点连通图ρG>θn,k, 然后αG>n-k2。界限是尖锐的,因为对于每个正整数nk+2,有图HρH=θn,kαG=n-k2

更新日期:2022-05-18
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