Skew Spectra of Oriented Bipartite Graphs

Abstract

A graph $G$ is said to have a parity-linked orientation $\phi$ if every even cycle $C_{2k}$ in $G^{\phi}$ is evenly (resp. oddly) oriented whenever $k$ 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 $G=G(X,Y)$ be a bipartite graph. Call the $X\rightarrow Y$ orientation of $G,$ the canonical orientation. Let $\phi$ be any orientation of $G$ and let $Sp_S(G^{\phi})$ and $Sp(G)$ denote respectively the skew spectrum of $G^{\phi}$ and the spectrum of $G.$ Then $Sp_S(G^{\phi}) = {\bf{i}} Sp(G)$ if and only if $\phi$ is switching-equivalent to the canonical orientation of $G.$ Using this result, we determine the switch for a special family of oriented hypercubes $Q_d^{\phi},$ $d\geq 1.$ 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.

5-regular oriented graphs with optimum skew energy

“…Now we find the four paths v 1 v 3 v 2 , v 1 v 4 v 2 , v 1 v 5 v 2 , v 1 v 6 v 2 of length 2 between v 1 and v 2 . Since w − v 1 v 2 (2) = 3, w + v 1 v 2 (2) = w − v 1 v 2 (2). Then this contradicts to Lemma 3.3.…”

A Note on Skew Energy of Hadamard Graph

“…In [8], Cui and Hou conjectured a necessary and sufficient condition for an oriented bipartite graph to have s as is switching equivalent to , where is the canonical orientation of . In [3], Anuradha et al settled down the conjecture as affirmative.…”

“…The skew spectrum of is defined as the spectrum of the skew adjacency matrix and consequently the skew energy is defined as the sum of the absolute values of its eigen values, denoted by . Skew energy was introduced by Adiga, Balakrishnan and So in [1] and was further studied by many authors [2], [3], [4], [8], [10], [14]. Now for every edge of Hadamard graph , it is natural to define an orientation according to the signs and as follows: if and if .…”

The skew-rank of oriented graphs

“…3. If n = 4, then G σ is one of the following oriented graphs with some properties: 3 and each edge has any orientation. (c) Evenly-oriented graph K σ 1,1,2 .…”

“…Anuradha et. al [3] considered the skew spectrum of special bipartite graphs and solved a conjecture of Cui and Hou [7]. Hou et al [9] gave an expression of the coefficients of the characteristic polynomial of the skew-adjacency matrix S(G σ ).…”