4 research outputs found

    Relation between the skew-rank of an oriented graph and the independence number of its underlying graph

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    An oriented graph GσG^\sigma is a digraph without loops or multiple arcs whose underlying graph is GG. Let S(Gσ)S\left(G^\sigma\right) be the skew-adjacency matrix of GσG^\sigma and α(G)\alpha(G) be the independence number of GG. The rank of S(Gσ)S(G^\sigma) is called the skew-rank of GσG^\sigma, denoted by sr(Gσ)sr(G^\sigma). Wong et al. [European J. Combin. 54 (2016) 76-86] studied the relationship between the skew-rank of an oriented graph and the rank of its underlying graph. In this paper, the correlation involving the skew-rank, the independence number, and some other parameters are considered. First we show that sr(Gσ)+2α(G)⩾2∣VG∣−2d(G)sr(G^\sigma)+2\alpha(G)\geqslant 2|V_G|-2d(G), where ∣VG∣|V_G| is the order of GG and d(G)d(G) is the dimension of cycle space of GG. We also obtain sharp lower bounds for sr(Gσ)+α(G), sr(Gσ)−α(G)sr(G^\sigma)+\alpha(G),\, sr(G^\sigma)-\alpha(G), sr(Gσ)/α(G)sr(G^\sigma)/\alpha(G) and characterize all corresponding extremal graphs.Comment: 16 Page; 1 figur

    Complex unit gain bicyclic graphs with rank 2, 3 or 4

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    A T\mathbb{T}-gain graph is a triple Φ=(G,T,φ)\Phi=(G,\mathbb{T},\varphi) consisting of a graph G=(V,E)G=(V,E), the circle group T={z∈C:∣z∣=1}\mathbb{T}=\{z\in C: |z|=1\} and a gain function φ:E→→T\varphi:\overrightarrow{E}\rightarrow \mathbb{T} such that φ(eij)=φ(eji)−1=φ(eji)‾\varphi(e_{ij})=\varphi(e_{ji})^{-1}=\overline{\varphi(e_{ji})}. The rank of T\mathbb{T}-gain graph Φ\Phi, denoted by r(Φ)r(\Phi), is the rank of the adjacency matrix of Φ\Phi. In 2015, Yu, Qu and Tu [ G. H. Yu, H. Qu, J. H. Tu, Inertia of complex unit gain graphs, Appl. Math. Comput. 265(2015) 619--629 ] obtained some properties of inertia of a T\mathbb{T}-gain graph. They characterized the T\mathbb{T}-gain unicyclic graphs with small positive or negative index. Motivated by above, in this paper, we characterize the complex unit gain bicyclic graphs with rank 2, 3 or 4.Comment: 15 pages, 4 figure

    The rank of a complex unit gain graph in terms of the rank of its underlying graph

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    Let Φ=(G,φ)\Phi=(G, \varphi) be a complex unit gain graph (or T\mathbb{T}-gain graph) and A(Φ)A(\Phi) be its adjacency matrix, where GG is called the underlying graph of Φ\Phi. The rank of Φ\Phi, denoted by r(Φ)r(\Phi), is the rank of A(Φ)A(\Phi). Denote by θ(G)=∣E(G)∣−∣V(G)∣+ω(G)\theta(G)=|E(G)|-|V(G)|+\omega(G) the dimension of cycle spaces of GG, where ∣E(G)∣|E(G)|, ∣V(G)∣|V(G)| and ω(G)\omega(G) are the number of edges, the number of vertices and the number of connected components of GG, respectively. In this paper, we investigate bounds for r(Φ)r(\Phi) in terms of r(G)r(G), that is, r(G)−2θ(G)≤r(Φ)≤r(G)+2θ(G)r(G)-2\theta(G)\leq r(\Phi)\leq r(G)+2\theta(G), where r(G)r(G) is the rank of GG. As an application, we also prove that 1−θ(G)≤r(Φ)r(G)≤1+θ(G)1-\theta(G)\leq\frac{r(\Phi)}{r(G)}\leq1+\theta(G). All corresponding extremal graphs are characterized.Comment: 17 pages. arXiv admin note: text overlap with arXiv:1612.0504

    Mixed graphs with cut vertices having exactly two positive eigenvalues

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    A mixed graph is obtained by orienting some edges of a simple graph. The positive inertia index of a mixed graph is defined as the number of positive eigenvalues of its Hermitian adjacency matrix, including multiplicities. This matrix was introduced by Liu and Li, independently by Guo and Mohar, in the study of graph energy. Recently, Yuan et al. characterized the mixed graphs with exactly one positive eigenvalue. In this paper, we study the positive inertia indices of mixed graphs and characterize the mixed graphs with cut vertices having positive inertia index 2.Comment: 26 pages, 4 figure
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