Geodesic
Solution to a conjecture on the maximum skew-spectral radius of odd-cycle graphs
Xiaolin Chen ; Xueliang Li1 ; Huishu Lian
1Nankai University
The electronic journal of combinatorics, Tome 22 (2015) no. 1
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Let $G$ be a simple graph with no even cycle, called an odd-cycle graph. Cavers et al. [Linear Algebra Appl. 436(12):4512-1829, 2012] showed that the spectral radius of $G^\sigma$ is the same for every orientation $\sigma$ of $G$, and equals the maximum matching root of $G$. They proposed a conjecture that the graphs which attain the maximum skew spectral radius among the odd-cycle graphs $G$ of order $n$ are isomorphic to the odd-cycle graph with one vertex degree $n-1$ and size $m=\lfloor 3(n-1)/2\rfloor$. By using the Kelmans transformation, we give a proof to the conjecture. Moreover, sharp upper bounds of the maximum matching roots of the odd-cycle graphs with given order $n$ and size $m$ are given and extremal graphs are characterized.
DOI : 10.37236/4919
Classification : 05C20, 05C50, 05C90
Mots-clés : skew spectral radius, odd-cycle graphs, maximum matching root, Kelmans transformation

Xiaolin Chen    ; Xueliang Li  1   ; Huishu Lian 

1 Nankai University 
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Xiaolin Chen; Xueliang Li; Huishu Lian. Solution to a conjecture on the maximum skew-spectral radius of odd-cycle graphs. The electronic journal of combinatorics, Tome 22 (2015) no. 1. doi: 10.37236/4919

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