A note on the nullity of unicyclic signed graphs

Fan, Yi-Zheng; Wang, Yue; Wang, Yi

doi:10.1016/j.laa.2012.08.027

Cited by 40 publications

(34 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section we shall consider the same question following the ideas in [12]. In Section 3, we characterized the weighted unicyclic graphs of order n with rank 4.…”

Section: Similar To the Above Result We Havementioning

confidence: 99%

The inertia of weighted unicyclic graphs

Yu¹,

Zhang²,

Feng³

2014

Linear Algebra and its Applications

View full text Add to dashboard Cite

Let G w be a weighted graph. The inertia of G w is the triple In(are the number of the positive, negative and zero eigenvalues of the adjacency matrix A(G w ) of G w including their multiplicities, respectively. i + (G w ), i − (G w ) is called the positive, negative index of inertia of G w , respectively. In this paper we present a lower bound for the positive, negative index of weighted unicyclic graphs of order n with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound. Moreover, we characterize the weighted unicyclic graphs of order n with two positive, two negative and at least n − 6 zero eigenvalues, respectively.

show abstract

“…In this section we shall consider the same question following the ideas in [12]. In Section 3, we characterized the weighted unicyclic graphs of order n with rank 4.…”

Section: Similar To the Above Result We Havementioning

confidence: 99%

The inertia of weighted unicyclic graphs

Yu¹,

Zhang²,

Feng³

2014

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…(2) ( [10]) Let Γ(C n ) be an unbalanced signed cycle. Then η(Γ(C n )) = 2 if n ≡ 2( mod 4) and η(Γ(C n )) = 0 otherwise.…”

Section: The Coefficients Of P γ(G) (λ)mentioning

confidence: 99%

Further Results on the Nullity of Signed Graphs

Liu

You

2014

Journal of Applied Mathematics

View full text Add to dashboard Cite

The nullity of a graph is the multiplicity of the eigenvalues zero in its spectrum. A signed graph is a graph with a sign attached to each of its edges. In this paper, we obtain the coefficient theorem of the characteristic polynomial of a signed graph, give two formulae on the nullity of signed graphs with cut-points. As applications of the above results, we investigate the nullity of the bicyclic signed graph Γ(∞(p, q, l)), obtain the nullity set of unbalanced bicyclic signed graphs, and thus determine the nullity set of bicyclic signed graphs.

show abstract

“…Guo et al [6] and Liu et al [12] introduced the Hermitian adjacency matrix of a mixed graph and presented some basic properties of the rank of the mixed graphs independently. In [4], the rank of the signed unicyclic graph was discussed by Fan et al He et al characterized the relation among the rank, the matching number and the cyclomatic number of a signed graph in [7]. Chen et al [3] investigated the relation between the H-rank of a mixed graph and the matching number of its underlying graph.…”

Section: Introductionmentioning

confidence: 99%

The rank of a complex unit gain graph in terms of the matching number

Hao

Dong

2020

Linear Algebra and its Applications

View full text Add to dashboard Cite

A complex unit gain graph (or T-gain graph) is a triple Φ = (G, T, ϕ) ((G, ϕ) for short) consisting of a graph G as the underlying graph of (G, ϕ), T = {z ∈ C : |z| = 1} is a subgroup of the multiplicative group of all nonzero complex numbers C × and a gain function ϕ :− → E → T such that ϕ(e ij ) = ϕ(e ji ) −1 = ϕ(e ji ). In this paper, we investigate the relation among the rank, the independence number and the cyclomatic number of a complex unit gain graph (G, ϕ) with order n, and prove that 2n − 2c(G) ≤ r(G, ϕ) + 2α(G) ≤ 2n. Where r(G, ϕ), α(G) and c(G) are the rank of the Hermitian adjacency matrix A(G, ϕ), the independence number and the cyclomatic number of G, respectively. Furthermore, the properties of the complex unit gain graph that reaching the lower bound are characterized.

show abstract

A note on the nullity of unicyclic signed graphs

Abstract: In this paper we introduce the nullity of signed graphs, and give some results on the nullity of signed graphs with pendant trees. We characterize the unicyclic signed graphs of order n with nullity n − 2, n − 3, n − 4, n − 5 respectively.

Cited by 40 publications

References 28 publications

The inertia of weighted unicyclic graphs

The inertia of weighted unicyclic graphs

Further Results on the Nullity of Signed Graphs

The rank of a complex unit gain graph in terms of the matching number

Contact Info

Product

Resources

About