Complex Unit Gain Graph Research Articles

Spectral Properties of Dual Unit Gain Graphs

In this paper, we study dual quaternion, dual complex unit gain graphs, and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications in robotics and computer graphics. Dual complex numbers have found application in brain science recently. We establish the interlacing theorem for dual unit gain graphs, and show that the spectral radius of a dual unit gain graph is always not greater than the spectral radius of the underlying graph, and these two radii are equal if, and only if, the dual gain graph is balanced. By using dual cosine functions, we establish the closed form of the eigenvalues of adjacency and Laplacian matrices of dual complex and quaternion unit gain cycles. We then show the coefficient theorem holds for dual unit gain graphs. Similar results hold for the spectral radius of the Laplacian matrix of the dual unit gain graph too.

Bounds of nullity for complex unit gain graphs

A complex unit gain graph, or T-gain graph, is a triple Φ=(G,T,φ) comprised of a simple graph G as the underlying graph of Φ, the set of unit complex numbers T={z∈C:|z|=1}, and a gain function φ:E→→T with the property that φ(eij)=φ(eji)−1. A cactus graph is a connected graph in which any two cycles have at most one vertex in common.In this paper, we firstly show that there does not exist a complex unit gain graph with nullity n(G)−2m(G)+2c(G)−1, where n(G), m(G) and c(G) are the order, matching number, and cyclomatic number of G. Next, we provide a lower bound on the nullity for connected complex unit gain graphs and an upper bound on the nullity for complex unit gain bipartite graphs. Finally, we characterize all non-singular complex unit gain bipartite cactus graphs, which generalizes a result in Wong et al. (2022) [30].

Eigenvalues of complex unit gain graphs and gain regularity

Abstract A complex unit gain graph (or T {\mathbb{T}} -gain graph) Γ = ( G , γ ) \Gamma =\left(G,\gamma ) is a gain graph with gains in T {\mathbb{T}} , the multiplicative group of complex units. The T {\mathbb{T}} -outgain in Γ \Gamma of a vertex v ∈ G v\in G is the sum of the gains of all the arcs originating in v v . A T {\mathbb{T}} -gain graph is said to be an a a - T {\mathbb{T}} -regular graph if the T {\mathbb{T}} -outgain of each of its vertices is equal to a a . In this article, it is proved that a a - T {\mathbb{T}} -regular graphs exist for every a ∈ R a\in {\mathbb{R}} . This, in particular, means that every real number can be a T {\mathbb{T}} -gain graph eigenvalue. Moreover, denoted by Ω ( a ) \Omega \left(a) the class of connected T {\mathbb{T}} -gain graphs whose largest eigenvalue is the real number a a , it is shown that Ω ( a ) \Omega \left(a) is nonempty if and only if a a belongs to { 0 } ∪ [ 1 , + ∞ ) \left\{0\right\}\cup \left[1,+\infty ) . In order to achieve these results, non-complete extended p p -sums and suitably defined joins of T {\mathbb{T}} -gain graphs are considered.

On connected [formula omitted]-gain graphs with rank equal to girth

Let Φ=(Γ,φ) be a T-gain graph or a complex unit gain graph and A(Φ) be its adjacency matrix. The rank of (Γ,φ) is denoted by r(Γ,φ) which is the rank of its adjacency matrix. If the underlying graph Γ of Φ has at least one cycle, then the girth of Φ is denoted by gr(Γ) or simply by gr, which is the length of the shortest cycle in Γ. In this paper, we prove r(Γ,φ)≥gr−2 for a T-gain graph (Γ,φ). Moreover, we characterize T-gain graphs satisfy r(Γ,φ)=gr−2 and r(Γ,φ)=gr.

NEPS of complex unit gain graphs

A complex unit gain graph (or $\mathbb T$-gain graph) is a gain graph with gains in $\mathbb T$, the multiplicative group of complex units. Extending a classical construction for simple graphs due to Cvektovic, suitably defined noncomplete extended $p$-sums (NEPS, for short) of $\mathbb T$-gain graphs are considered in this paper. Structural properties of NEPS like balance and some spectral properties and invariants of their adjacency and Laplacian matrices are investigated, including the energy and the possible symmetry of the adjacency spectrum. It is also shown how NEPS are useful to obtain infinitely many integral graphs from the few at hands.Moreover, it is studied how NEPS of $\mathbb T$-gain graphs behave with respect to the property of being nut, i.e., having $0$ as simple adjacency eigenvalue and nowhere zero $0$-eigenvectors. Finally, a family of new products generalizing NEPS is introduced, and their few first spectral properties explored.

On bounds of Aα-eigenvalue multiplicity and the rank of a complex unit gain graph

Let Φ=(G,φ) be a connected complex unit gain graph (T-gain graph) of n vertices with largest vertex degree Δ, adjacency matrix A(Φ), and degree matrix D(Φ). Let mα(Φ,λ) be the multiplicity of λ as an eigenvalue of Aα(Φ):=αD(Φ)+(1−α)A(Φ), for α∈[0,1). In this article, we establish that mα(Φ,λ)≤(Δ−2)n+2Δ−1 and characterize the sharpness. Then, we obtain some lower bounds for the rank r(Φ) in terms of n and Δ including r(Φ)≥n−2Δ−1 and characterize their sharpness. Besides, we introduce zero-2-walk gain graphs and study their properties. It is shown that a zero-2-walk gain graph is always regular. Furthermore, we prove that Φ has exactly two distinct eigenvalues with equal magnitude if and only if it is a zero-2-walk gain graph. Using this, we establish a lower bound of r(Φ) in terms of the number of edges and characterize the sharpness. Result about mα(Φ,λ) extends the corresponding known result for undirected graphs and simplifies the existing proof, and other bounds of r(Φ) obtained in this article work better than the bounds given elsewhere.

Relation between the row left rank of a quaternion unit gain graph and the rank of its underlying graph

Let $\Phi=(G,U(\mathbb{Q}),\varphi)$ be a quaternion unit gain graph (or $U(\mathbb{Q})$-gain graph), where $G$ is the underlying graph of $\Phi$, $U(\mathbb{Q})=\{z\in \mathbb{Q}: |z|=1\}$ is the circle group, and $\varphi:\overrightarrow{E}\rightarrow U(\mathbb{Q})$ is the gain function such that $\varphi(e_{ij})=\varphi(e_{ji})^{-1}=\overline{\varphi(e_{ji})}$. Let $A(\Phi)$ be the adjacency matrix of $\Phi$ and $r(\Phi)$ be the row left rank of $\Phi$. In this paper, we prove that $-2c(G)\leq r(\Phi)-r(G)\leq 2c(G)$, where $r(G)$ and $c(G)$ are the rank and the dimension of cycle space of $G$, respectively. All corresponding extremal graphs are characterized. The results will generalize the corresponding results of signed graphs (Lu et al. [20] and Wang [33]), mixed graphs (Chen et al. [7]), and complex unit gain graphs (Lu et al. [21]).

Inertia indices of a complex unit gain graph in terms of matching number

A complex unit gain graph is a triple (or for short) consisting of a simple graph G, as the underlying graph of , the set of unit complex numbers and a gain function such that . Let be the adjacency matrix of . In this paper, we prove that where , , and are the number of positive eigenvalues of , the number of negative eigenvalues of , the matching number and the cyclomatic number of G, respectively. Furthermore, we characterize the graphs which attain the upper bounds and the lower bounds, respectively.

Unit gain graphs with two distinct eigenvalues and systems of lines in complex space

Since the introduction of the Hermitian adjacency matrix for digraphs, interest in so-called complex unit gain graphs has surged. In this work, we consider gain graphs whose spectra contain the minimum number of two distinct eigenvalues. Analogously to graphs with few distinct eigenvalues, a great deal of structural symmetry is required for a gain graph to attain this minimum. This allows us to draw a surprising parallel to well-studied systems of lines in complex space, through a natural correspondence to unit-norm tight frames. We offer a full classification of two-eigenvalue gain graphs with degree at most 4, or with multiplicity at most 3. Intermediate results include an extensive review of various relevant concepts related to lines in complex space, including SIC-POVMs, MUBs and geometries such as the Coxeter-Todd lattice, and many examples obtained as induced subgraphs by employing a technique parallel to the dismantling of association schemes. Finally, we touch on an innovative application of simulated annealing to find examples by computer.

Gain distance matrices for complex unit gain graphs

A complex unit gain graph (T-gain graph) Φ=(G,φ) is a graph where the function φ assigns a unit complex number to each orientation of an edge of G, and its inverse is assigned to the opposite orientation. A T-gain graph is balanced if the product of the edge gains of each oriented cycle (if any) is 1. We propose two notions of gain distance matrices D<max(Φ) and D<min(Φ) of a T-gain graph Φ, for any ordering ‘<’ of the vertex set. We characterize the gain graphs for which the gain distance matrices are independent of the vertex ordering. We show D<max(Φ)=D<min(Φ) holds for the standard ordering of the vertices if and only if the same holds for any ordering of the vertices, and we call such T-gain graphs as distance compatible gain graphs. We characterize the distance compatible gain graphs whose gain distance matrices are cospectral with the distance matrix of the underlying graph. Besides, we introduce the notion of positively weighted T-gain graphs and establish an equivalent condition for the balance of a T-gain graph. Acharya's and Stanić's spectral criteria for balance are deduced as a consequence. Besides, we obtain some spectral characterizations for the balance of a T-gain graph in terms of the gain distance matrices. Finally, we characterize the distance compatible bipartite T-gain graphs. We show a T-gain graph Φ is distance compatible if and only if every block of Φ is distance compatible.

Spectra of quaternion unit gain graphs

A quaternion unit gain graph is a graph where each orientation of an edge is given a quaternion unit, which is the inverse of the quaternion unit assigned to the opposite orientation. In this paper we define the adjacency, Laplacian and incidence matrices for a quaternion unit gain graph and study their properties. These properties generalize several fundamental results from spectral graph theory of ordinary graphs, signed graphs and complex unit gain graphs. Bounds for both the left and right eigenvalues of the adjacency and Laplacian matrix are developed, and the right eigenvalues for the cycle and path graphs are explicitly calculated.

Bounds for the extremal eigenvalues of gain Laplacian matrices

A complex unit gain graph (T-gain graph), Φ=(G,φ) is a graph where the function φ assigns a unit complex number to each orientation of an edge of G, and its inverse is assigned to the opposite orientation. A T-gain graph Φ is balanced if the product of the edge gains of each cycle (with a fixed orientation) is 1. Signed graphs are special cases of T-gain graphs.The adjacency matrix of Φ, denoted by A(Φ) is defined canonically. The gain Laplacian for Φ is defined as L(Φ)=D(Φ)−A(Φ), where D(Φ) is the diagonal matrix with diagonal entries are the degrees of the vertices of G. The minimum number of vertices (resp., edges) to be deleted from Φ in order to get a balanced gain graph is the frustration number (resp, frustration index). We show that frustration number and frustration index are bounded below by the smallest eigenvalue of L(Φ). We provide several lower and upper bounds for extremal eigenvalues of L(Φ) in terms of different graph parameters such as the number of edges, vertex degrees, and average 2-degrees. The signed graphs are particular cases of the T-gain graphs, all the bounds established in this paper hold for signed graphs. Most of the bounds established here are new for signed graphs. Finally, we perform comparative analysis for all the obtained bounds in the paper with the state-of-the-art bounds available in the literature for randomly generated Erdős-Reýni graphs.Some of the major highlights of our paper are the gain-dependent bounds, limit convergence of the bounds to the extremal eigenvalues, and optimal extremal bounds obtained by posing optimization problems to achieve the best possible bounds.

Line graphs of complex unit gain graphs with least eigenvalue -2

Let $\mathbb T$ be the multiplicative group of complex units, and let $\mathcal L (\Phi)$ denote a line graph of a $\mathbb{T}$-gain graph $\Phi$. Similarly to what happens in the context of signed graphs, the real number $\min Spec(A(\mathcal L (\Phi))$, that is, the smallest eigenvalue of the adjacency matrix of $\mathcal L(\Phi)$, is not less than $-2$. The structural conditions on $\Phi$ ensuring that $\min Spec(A(\mathcal L (\Phi))=-2$ are identified. When such conditions are fulfilled, bases of the $-2$-eigenspace are constructed with the aid of the star complement technique.

Bounds for the energy of a complex unit gain graph

A T-gain graph, Φ=(G,φ), is a graph in which the function φ assigns a unit complex number to each orientation of an edge of G, and its inverse is assigned to the opposite orientation. The associated adjacency matrix A(Φ) is defined canonically. The energy E(Φ) of a T-gain graph Φ is the sum of the absolute values of all eigenvalues of A(Φ). We study the notion of energy of a vertex of a T-gain graph, and establish bounds for it. For any T-gain graph Φ, we prove that 2τ(G)−2c(G)≤E(Φ)≤2τ(G)Δ(G), where τ(G),c(G) and Δ(G) are the vertex cover number, the number of odd cycles and the largest vertex degree of G, respectively. Furthermore, using the properties of vertex energy, we characterize the class of T-gain graphs for which E(Φ)=2τ(G)−2c(G) holds. Also, we characterize the T-gain graphs for which E(Φ)=2τ(G)Δ(G) holds. This characterization solves a general version of an open problem. In addition, we establish bounds for the energy in terms of the spectral radius of the associated adjacency matrix.

Bounds for the rank of a complex unit gain graph in terms of its maximum degree

Let Φ=(G,φ) be a T-gain graph with maximum degree Δ. Zhou et al. (2018) [35] gave a lower bound of the rank of simple graphs in items of maximum degree. Motivated by above result, in this paper, we extend this result to T-gain graphs. We obtain that r(Φ)≥nΔ for a T-gain graph. All the corresponding extremal graphs are characterized. Furthermore, we also obtain some other results about the rank of T-gain graphs.

Complex unit gain graphs with exactly one positive eigenvalue

A complex unit gain graph is a graph where each orientation of an edge is given a complex unit, which is the inverse of the complex unit assigned to the opposite orientation. In this paper, we characterize the structure of the complex unit gain graphs with exactly one positive eigenvalue. As its applications, we obtain the complex unit gain graphs with rank 2, and investigate the complex unit gain graphs with exactly two eigenvalues different from 0 and −1.

On the adjacency matrix of a complex unit gain graph

ABSTRACT A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for the eigenvalues of the complex unit gain graphs. Then we study some of the properties of the adjacency matrix of a complex unit gain graph in connection with the characteristic and the permanental polynomials. Then we establish spectral properties of the adjacency matrices of complex unit gain graphs. In particular, using Perron–Frobenius theory, we establish a characterization for bipartite graphs in terms of the set of eigenvalues of a gain graph and the set of eigenvalues of the underlying graph. Also, we derive an equivalent condition on the gain so that the eigenvalues of the gain graph and the eigenvalues of the underlying graph are the same.

On the relation between the adjacency rank of a complex unit gain graph and the matching number of its underlying graph

ABSTRACT Let be an n-vertex complex unit gain graph and let G be its underlying graph. The adjacency rank of , written as , is the rank of its adjacency matrix and denote by the matching number of the underlying graph G. In this contribution, based on combinatorial interpretation of all the coefficients of the characteristic polynomial of , we determine sharp upper and lower bounds on Furthermore, we establish sharp lower bounds on and . All the corresponding extremal complex unit gain graphs are characterized.

Relation between the inertia indices of a complex unit gain graph and those of its underlying graph

A -gain graph is a triple consisting of an underlying graph , the circle group and a gain function , such that . In this paper, we focus our attention on the relations between the inertia indices of -gain graph Φ and the inertia indices of its underlying graph G. We obtain sharp lower and upper bounds on (resp., ) in terms of (resp., ) and characterize those corresponding extremal -gain graphs, respectively. As a corollary, we also characterize those signed graphs whose inertia indices obtaining the sharp bounds, respectively.

Bounds for the rank of a complex unit gain graph in terms of the independence number

A complex unit gain graph (or -gain graph) is a triple (or for short) consisting of a simple graph G with , as the underlying graph of , the set of unit complex numbers and a gain function with the property that . The adjacency matrix of is , where if is adjacent to and otherwise. The rank of , denoted by , is the rank of . Let and be the independence number and the cyclomatic number of G, respectively. In this paper, we prove that . And the properties of the complex unit gain graphs that attain the lower bound are characterized. Furthermore, the lower and upper bounds on , and are identified. These results generalize the corresponding known results about undirected graphs, mixed graphs, oriented graphs and signed graphs.