Skew Spectra of Oriented Bipartite Graphs (original) (raw)

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Keywords: oriented bipartite graphs, skew energy, skew spectrum, canonical orientation, parity-linked orientation, switching-equivalence

Abstract

A graph GGG is said to have a parity-linked orientation phi\phiphi if every even cycle C2kC_{2k}C2k in GphiG^{\phi}Gphi is evenly (resp. oddly) oriented whenever kkk is even (resp. odd). In this paper, this concept is used to provide an affirmative answer to the following conjecture of D. Cui and Y. Hou [D. Cui, Y. Hou, On the skew spectra of Cartesian products of graphs, Electronic J. Combin. 20(2):#P19, 2013]: Let G=G(X,Y)G=G(X,Y)G=G(X,Y) be a bipartite graph. Call the XrightarrowYX\rightarrow YXrightarrowY orientation of G,G,G, the canonical orientation. Let phi\phiphi be any orientation of GGG and let SpS(Gphi)Sp_S(G^{\phi})SpS(Gphi) and Sp(G)Sp(G)Sp(G) denote respectively the skew spectrum of GphiG^{\phi}Gphi and the spectrum of G.G.G. Then SpS(Gphi)=bfiSp(G)Sp_S(G^{\phi}) = {\bf{i}} Sp(G)SpS(Gphi)=bfiSp(G) if and only if phi\phiphi is switching-equivalent to the canonical orientation of G.G.G. Using this result, we determine the switch for a special family of oriented hypercubes Qdphi,Q_d^{\phi},Qdphi, dgeq1.d\geq 1.dgeq1. Moreover, we give an orientation of the Cartesian product of a bipartite graph and a graph, and then determine the skew spectrum of the resulting oriented product graph, which generalizes a result of Cui and Hou. Further this can be used to construct new families of oriented graphs with maximum skew energy.

How to Cite

Anuradha, A., Balakrishnan, R., Chen, X., Li, X., Lian, H., & So, W. (2013). Skew Spectra of Oriented Bipartite Graphs. The Electronic Journal of Combinatorics, 20(4), #P18. https://doi.org/10.37236/3331