Hermitian Adjacency Matrix Research Articles

Inverse of \(\alpha\)-Hermitian Adjacency Matrix of a Unicyclic Bipartite Graph

Let \(X\) be bipartite mixed graph and for a unit complex number \(\alpha\), \(H_\alpha\) be its \(\alpha\)-hermitian adjacency matrix. If \(X\) has a unique perfect matching, then \(H_\alpha\) has a hermitian inverse \(H_\alpha^{-1}\). In this paper we give a full description of the entries of \(H_\alpha^{-1}\) in terms of the paths between the vertices. Furthermore, for \(\alpha\) equals the primitive third root of unity \(\gamma\) and for a unicyclic bipartite graph \(X\) with unique perfect matching, we characterize when \(H_\gamma^{-1}\) is \(\pm 1\) diagonally similar to \(\gamma\)-hermitian adjacency matrix of a mixed graph. Through our work, we have provided a new construction for the \(\pm 1\) diagonal matrix.

The Index Weighted Hermitian Adjacency Matrices for Mixed Graphs

Based on the Hermitian adjacency matrices of second kind introduced by Mohar [1] and weighted adjacency matrices introduced in [2], we define a kind of index weighted Hermitian adjacency matrices of mixed graphs. In this paper we characterize the structure of mixed graphs which are cospectral to their underlying graphs, then we determine a upper bound on the spectral radius of mixed graphs with maximum degree Δ , and characterize the corresponding extremal graphs.

Quantum state transfer on integral oriented circulant graphs

An oriented circulant graph is called integral if all eigenvalues of its Hermitian adjacency matrix are integers. The main purpose of this paper is to investigate the existence of perfect state transfer (PST for short) and multiple state transfer (MST for short) on integral oriented circulant graphs. Specifically, a characterization of PST and MST on integral oriented circulant graphs is provided. As an application, we also obtain a closed-form expression for the number of integral oriented circulant graphs having PST and MST.

State transfer on integral mixed circulant graphs

A mixed circulant graph is called integral if all eigenvalues of its Hermitian adjacency matrix are integers. The main purpose of this paper is to investigate the existence of perfect state transfer (PST for short) and multiple state transfer (MST for short) on integral mixed circulant graphs. Concretely, we provide sufficient and necessary conditions for the existence of PST and MST between specified pairs of vertices on integral mixed circulant graphs, respectively.

H-integral and Gaussian integral normal mixed Cayley graphs

If all the eigenvalues of the Hermitian-adjacency matrix of a mixed graph are integers, then the mixed graph is called H-integral. If all the eigenvalues of the (0,1)-adjacency matrix of a mixed graph are Gaussian integers, then the mixed graph is called Gaussian integral. For any finite group Γ, we characterize the set S for which the normal mixed Cayley graph Cay(Γ,S) is H-integral. We further prove that a normal mixed Cayley graph is H-integral if and only if the mixed graph is Gaussian integral.

Mixed graphs whose Hermitian adjacency matrices of the second kind have the smallest eigenvalue greater than [formula omitted

Mixed graphs unify simple graphs and oriented graphs naturally. The Hermitian adjacency matrix of a mixed graph for the second kind (N-matrix for short) was recently introduced by Mohar [13]. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from u to v is equal to the principal sixth root of unity ω=1+i32 (and its symmetric entry is ω‾=1−i32); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. In this paper, we completely characterize the connected mixed graphs whose N-matrices have the smallest eigenvalue greater than −32. It may be considered as the continuance of the work of J.H. Smith [18] who considered the classical problem of characterizing the graphs whose eigenvalues lie in a given interval.

HS-splitting fields of abelian mixed cayley graphs

For an n-vertex mixed graph A, let HS(A) be the Hermitian-adjacency matrix of the second kind of A and ΦA(HS,λ)=det(λIn−HS(A)) the characteristic polynomial of HS(A). The splitting field of ΦA(HS,λ) is referred to as the HS-splitting field of A. Its extension degree over the rational number field Q is referred to as the HS-algebraic degree of A, and A is said to be HS-integral if all eigenvalues of HS(A) are integers. In this paper, we give explicit expressions for the HS-splitting fields of abelian mixed Cayley graphs. In addition, we derive a formula to calculate their corresponding HS-algebraic degrees. Moreover, we characterize all HS-integral abelian mixed Cayley graphs.

A spectral graph convolution for signed directed graphs via magnetic Laplacian

Signed directed graphs contain both sign and direction information on their edges, providing richer information about real-world phenomena compared to unsigned or undirected graphs. However, analyzing such graphs is more challenging due to their complexity, and the limited availability of existing methods. Consequently, despite their potential uses, signed directed graphs have received less research attention. In this paper, we propose a novel spectral graph convolution model that effectively captures the underlying patterns in signed directed graphs. To this end, we introduce a complex Hermitian adjacency matrix that can represent both sign and direction of edges using complex numbers. We then define a magnetic Laplacian matrix based on the adjacency matrix, which we use to perform spectral convolution. We demonstrate that the magnetic Laplacian matrix is positive semi-definite (PSD), which guarantees its applicability to spectral methods. Compared to traditional Laplacians, the magnetic Laplacian captures additional edge information, which makes it a more informative tool for graph analysis. By leveraging the information of signed directed edges, our method generates embeddings that are more representative of the underlying graph structure. Furthermore, we showed that the proposed method has wide applicability for various graph types and is the most generalized Laplacian form. We evaluate the effectiveness of the proposed model through extensive experiments on several real-world datasets. The results demonstrate that our method outperforms state-of-the-art techniques in signed directed graph embedding.

On the multiplicities of digraph eigenvalues

We show various upper bounds for the order of a digraph (or a mixed graph) whose Hermitian adjacency matrix has an eigenspace of prescribed codimension. In particular, this generalizes the so-called absolute bound for (simple) graphs first shown by Delsarte, Goethals, and Seidel (1977) and extended by Bell and Rowlinson (2003). In doing so, we also adapt the Blokhuis' theory (1983) of harmonic analysis in real hyperbolic spaces to that in complex hyperbolic spaces.

Pretty good state transfer in discrete-time quantum walks

We establish the theory for pretty good state transfer in discrete-time quantum walks. For a class of walks, we show that pretty good state transfer is characterized by the spectrum of certain Hermitian adjacency matrix of the graph; more specifically, the vertices involved in pretty good state transfer must be strongly cospectral relative to this matrix, and the arccosines of its eigenvalues must satisfy some number theoretic conditions. Using normalized adjacency matrices, cyclic covers, and the theory on linear relations between geodetic angles, we construct several infinite families of walks that exhibits this phenomenon.

HS-integral and Eisenstein integral normal mixed Cayley graphs

A mixed graph is said to be second kind hermitian integral (HS-integral) if the eigenvalues of its Hermitian-adjacency matrix of second kind are integers. A mixed graph is called Eisenstein integral if the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. We characterize the set S for which the normal mixed Cayley graph Cay(Γ,S) is HS-integral for any finite group Γ. We further show that a normal mixed Cayley graph is HS-integral if and only if it is Eisenstein integral. This paper generalizes the results of Kadyan and Bhattacharjy (2022) [11].

On the characteristic polynomial and energy of Hermitian quasi-Laplacian matrix of mixed graphs

A mixed graph is a graph whose edge set consists of both oriented and unoriented edges. The Hermitian-adjacency matrix of an [Formula: see text]-vertex mixed graph is a square matrix [Formula: see text] of order [Formula: see text], where [Formula: see text] if there is an arc from [Formula: see text] to [Formula: see text] and [Formula: see text] if there is an edge between [Formula: see text] and [Formula: see text], and [Formula: see text] otherwise. Let [Formula: see text] be a diagonal matrix, where [Formula: see text] is the degree of [Formula: see text] in the underlying graph of [Formula: see text]. The matrices [Formula: see text] and [Formula: see text] are, respectively, the Hermitian Laplacian and Hermitian quasi-Laplacian matrix of the mixed graph [Formula: see text]. In this paper, we first found coefficients of the characteristic polynomial of Hermitian Laplacian and Hermitian quasi-Laplacian matrices of the mixed graph [Formula: see text]. Second, we discussed relationship between the spectra of Hermitian Laplacian and Hermitian quasi-Laplacian matrices of the mixed graph [Formula: see text].

Why and how to add direction to a quantum walk

We formalize the treatment of directed (or chiral) quantum walks using Hermitian adjacency matrices, bridging two developing fields of research in quantum information and spectral graph theory. We display results and simulations which highlight the conceptual differences between having directions encoded in the Hamiltonians or not. This leads to a construction of a new type of quantum phenomenon: zero transfer between pairs of sites in a connected coupled network, which is only possible in the directed model we study. Our main result is a description of several families of directed cycles that admit zero transfer.

HS-integral and Eisenstein integral mixed circulant graphs

A mixed graph is called \emph{second kind hermitian integral} (\emph{HS-integral}) if the eigenvalues of its Hermitian-adjacency matrix of the second kind are integers. A mixed graph is called \emph{Eisenstein integral} if the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. We characterize the set $S$ for which a mixed circulant graph $\text{Circ}(\mathbb{Z}_n, S)$ is HS-integral. We also show that a mixed circulant graph is Eisenstein integral if and only if it is HS-integral. Further, we express the eigenvalues and the HS-eigenvalues of unitary oriented circulant graphs in terms of generalized M$\ddot{\text{o}}$bius function.

The k-generalized Hermitian adjacency matrices for mixed graphs

This article gives some fundamental introduction to spectra of mixed graphs via its k-generalized Hermitian adjacency matrix. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from u to v is equal to the kth root of unity e2πik (and its symmetric entry is e−2πik); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. For all positive integers k, the non-zero entries of the above matrix are chosen from the gain set {1,e2πik,e−2πik}, which is not closed under multiplication when k⩾4. In this paper, for all positive integers k, we extract all the mixed graphs whose k-generalized Hermitian adjacency rank (Hk-rank for short) is 3, which partially answers a question proposed by Wissing and van Dam [34]. Furthermore, we study the spectral determination of mixed graphs with Hk-rank 2 and 3, respectively.

Integral mixed circulant graphs

A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integers. The mixed circulant graphCirc(Zn,C) is a mixed graph on the vertex set Zn and edge set {(a,b):b−a∈C}, where 0∉C⊂Zn. If C is closed under inverse, then Circ(Zn,C) is called a circulant graph. We express the eigenvalues of Circ(Zn,C) in terms of primitive n-th roots of unity, and find a sufficient condition for integrality of the eigenvalues of Circ(Zn,C). For n≡0(mod4), we factorize the n-th degree cyclotomic polynomial into two irreducible factors over Q(i). Using this factorization, we characterize integral mixed circulant graphs in terms of their symbol set. We also express the integer eigenvalues of an integral oriented circulant graph in terms of a Ramanujan type sum, and discuss some of their properties.

Inverse of Hermitian Adjacency Matrix of a Mixed Graph

Inverse of Hermitian Adjacency Matrix of a Mixed Graph

The spectra of random mixed graphs

A mixed graph is a graph that can be obtained from a simple undirected graph by replacing some of the edges by arcs in precisely one of the two possible directions. The Hermitian adjacency matrix of a mixed graph G of order n is the n×n matrix H(G)=(hij), where hij=−hji=i (with i=−1) if there exists an arc from vi to vj (but no arc from vj to vi), hij=hji=1 if there exists an edge (and no arcs) between vi and vj, and hij=hji=0 otherwise (if vi and vj are neither joined by an edge nor by an arc). Let λ1(G),λ2(G),…,λn(G) be eigenvalues of H(G). The k-th Hermitian spectral moment of G is defined as sk(H(G))=∑i=1nλik(G), where k≥0 is an integer. In this paper, we deal with the asymptotic behavior of the spectrum of the Hermitian adjacency matrix of random mixed graphs. We will present and prove a separation result between the largest and remaining eigenvalues of the Hermitian adjacency matrix, and as an application, we estimate the Hermitian spectral moments of random mixed graphs.

Hermitian Adjacency Matrices of Mixed Graphs

The traditional adjacency matrix of a mixed graph is not symmetric in general, hence its eigenvalues may be not real. To overcome this obstacle, several authors have recently defined and studied various Hermitian adjacency matrices of digraphs or mixed graphs. In this work we unify previous work and offer a new perspective on the subject by introducing the concept of monographs. Moreover, we consider questions of cospectrality.

On the Cospectrality of Hermitian Adjacency Matrices of Mixed Graphs

A mixed graph D is a graph that can be obtained from a graph by orienting some of its edges. Let α be a primitive n th root of unity, then the α−Hermitian adjacency matrix of a mixed graph is defined to be the matrix Hα = [hrs] where hrs = α if rs is an arc in D, hrs = α if sr is an arc in D, hrs = 1 if sr is a digon in D and hrs = 0 otherwise. In this paper we study the cospectrality of the Hermitian adjacency matrix of a mixed graph.