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This contribution gives an extensive study on spectra of mixed graphs via its Hermitian adjacency matrix of the second kind { ($N$-matrix for short)} introduced by Mohar \cite{0001}. 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 sixth root of unity $\omega=\frac{1+{\bf i}\sqrt{3}}{2}$ (and its symmetric entry is $\bar{\omega}=\frac{1-{\bf i}\sqrt{3}}{2}$); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. The main results of this paper include the following: {equivalent} conditions for a mixed graph that shares the same spectrum of its $N$-matrix with its underlying graph are given. A sharp upper bound on the spectral radius is established and the corresponding extremal mixed graphs are identified. Operations which are called two-way and three-way switchings are discussed--they give rise to some cospectral mixed graphs. We extract all the mixed graphs whose rank of its $N$-matrix is $2$ (resp. 3). Furthermore, we show that {if $M_G$ is a connected mixed graph with rank $2,$ then $M_G$ is switching equivalent to each connected mixed graph to which it is cospectral}. However, this does not hold for some connected mixed graphs with rank $3$. We identify all mixed graphs whose eigenvalues of its $N$-matrix lie in the range $(-\alpha,\, \alpha)$ for $\alpha\in\left\{\sqrt{2},\,\sqrt{3},\,2\right\}$., Comment: 31pages,15figures

We describe how eight copies of the Best code can be imbedded in the binary Hamming graph of length 10 in such a way that they form part of a partition design π1. We study those isometries of the binary Hamming graph which stabilize π1. A partition design π2, coarser than π1, will be the subject of the second part of this work. Let M be the adjacency matrix of π2. We will show that M is extreme in some sense. Namely, π2 has the following interesting property: π2 is the unique partition design with adjacency matrix M.

Let G be a graph without loops and multiple edges. If V(G) = (V[in1]), V2, …, vn, we define the adjacency matrix of G to be the n × n (0, I)-matrix A (G) = (aij), where aij = 1 if vivjϵ E (G) and aij = 0 otherwise. G is said to be singular if the matrix A(G) is singular. Reduction procedures which will decrease the amount of computation needed to obtain the determinant of the adjacency matrices of some graphs are introduced. One of these reduction procedures is used in proving the singularity of square planar grid Pn × Pn.

Recently, several results bounding above the diameter and/or the mean distance of a graph from its eigenvalues have been presented. They use the eigenvalues of either the adjacency or the Laplacian matrix of the graph. The main object of this paper is to compare both methods. As expected, they are equivalent for regular graphs. However, the situation is different for nonregular graphs: While no method has a definite advantage when bounding above the diameter, the use of the Laplacian matrix seems better when dealing with the mean distance. This last statement follows from improved bounds on the mean distance obtained in the paper.

We present a sequence of graphs G n for which χ(G n ) ⩾ Ω( rk (A(G n )) 4 3 ) .

“Which graphs are determined by their spectrum (DS for short)?” is a fundamental question in spectral graph theory. It is generally very hard to show a given graph to be DS and few results about DS graphs are known in literature. In this paper, we consider the above problem in the context of the generalized Q -spectrum. A graph G is said to be determined by the generalized Q -spectrum (DGQS for short) if, for any graph H , H and G have the same Q -spectrum and so do their complements imply that H is isomorphic to G . We give a simple arithmetic condition for a graph being DGQS. More precisely, let G be a graph with adjacency matrix A and degree diagonal matrix D . Let Q = A + D be the signless Laplacian matrix of G , and W Q ( G ) = [ e , Q e , … , Q n − 1 e ] ( e is the all-ones vector) be the Q -walk matrix. We show that if det W Q ( G ) 2 ⌊ 3 n − 2 2 ⌋ (which is always an integer) is odd and square-free, then G is DGQS.

It is well-known that the paths are determined by the spectrum of the adjacency matrix. For digraphs, every digraph whose underlying graph is a tree is cospectral to its underlying graph with respect to the Hermitian adjacency matrix ( H -cospectral). Thus every (simple) digraph whose underlying graph is isomorphic to P n is H -cospectral to P n . Interestingly, there are others. This paper finds digraphs that are H -cospectral with the path graph P n and whose underlying graphs are nonisomorphic, when n is odd, and finds that such graphs do not exist when n is even. In order to prove this result, all digraphs whose Hermitian spectral radius is smaller than 2 are determined.

Let G be a simple connected graph with n vertices and m edges. The spectral radius ρ ( G ) of G is the largest eigenvalue of its adjacency matrix. In this paper, we firstly consider the effect on the spectral radius of a graph by removing a vertex, and then as an application of the result, we obtain a new sharp upper bound of ρ ( G ) which improves some known bounds: If ( k − 2 ) ( k − 3 ) 2 ≤ m − n ≤ k ( k − 3 ) 2 , where k ( 3 ≤ k ≤ n ) is an integer, then ρ ( G ) ≤ 2 m − n − k + 5 2 + 2 m − 2 n + 9 4 . The equality holds if and only if G is a complete graph K n or K 4 − e , where K 4 − e is the graph obtained from K 4 by deleting some edge e .

A signed graph is a pair Γ = ( G , σ ) , where x = ( V ( G ) , E ( G ) ) is a graph and σ : E ( G ) → { + 1 , − 1 } is the sign function on the edges of G . For any α ∈ [ 0 , 1 ] we consider the matrix A α ( Γ ) = α D ( G ) + ( 1 − α ) A ( Γ ) , where D ( G ) is the diagonal matrix of the vertex degrees of G , and A ( Γ ) is the adjacency matrix of Γ . Let m A α ( Γ ) ( α ) be the multiplicity of α as an A α ( Γ ) -eigenvalue, and let G have p ( G ) pendant vertices, q ( G ) quasi-pendant vertices, and no components isomorphic to K 2 . It is proved that m A α ( Γ ) ( α ) = p ( G ) − q ( G ) whenever all internal vertices are quasi-pendant. If this is not the case, it turns out that m A α ( Γ ) ( α ) = p ( G ) − q ( G ) + m N α ( Γ ) ( α ) , where m N ( Γ ) ( α ) denotes the multiplicity of α as an eigenvalue of the matrix N α ( Γ ) obtained from A α ( Γ ) taking the entries corresponding to the internal vertices which are not quasi-pendant. Such results allow to state a formula for the multiplicity of 1 as an eigenvalue of the Laplacian matrix L ( Γ ) = D ( G ) − A ( Γ ) . Furthermore, it is detected a class G of signed graphs whose nullity – i.e. the multiplicity of 0 as an A ( Γ ) -eigenvalue – does not depend on the chosen signature. The class G contains, among others, all signed trees and all signed lollipop graphs. It also turns out that for signed graphs belonging to a subclass G ′ ⊂ G the multiplicity of 1 as Laplacian eigenvalue does not depend on the chosen signatures. Such subclass contains trees and circular caterpillars.

We study functions defined on the vertices of the Hamming graphs $H(n,q)$. The adjacency matrix of $H(n,q)$ has $n+1$ distinct eigenvalues $n(q-1)-q\cdot i$ with corresponding eigenspaces $U_{i}(n,q)$ for $0\leq i\leq n$. In this work, we consider the problem of finding the minimum possible support (the number of nonzeros) of functions belonging to a direct sum $U_i(n,q)\oplus U_{i+1}(n,q)\oplus\ldots\oplus U_j(n,q)$ for $0\leq i\leq j\leq n$. For the case $n\geq i+j$ and $q\geq 3$ we find the minimum cardinality of the support of such functions and obtain a characterization of functions with the minimum cardinality of the support. In the case $n \frac{n}{2}$,$\,q\ge 5$., Comment: 17 pages, 3 figures

Let G be a graph with n vertices, and let A ( G ) and D ( G ) denote respectively the adjacency matrix and the degree matrix of G . Define A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) for any real α ∈ [ 0 , 1 ] . The collection of eigenvalues of A α ( G ) together with multiplicities are called the A α -spectrum of G . A graph G is said to be determined by its A α -spectrum if all graphs having the same A α -spectrum as G are isomorphic to G . We first prove that some graphs are determined by their A α -spectra for 0 ≤ α 1 , including the complete graph K n , the union of cycles, the complement of the union of cycles, the union of copies of K 2 and K 1 , the complement of the union of copies of K 2 and K 1 , the path P n , and the complement of P n . Setting α = 0 or 1 2 , those graphs are determined by A - or Q -spectra. Secondly, when G is regular, we show that G is determined by its A α -spectrum if and only if the join G ∨ K m ( m ≥ 2 ) is determined by its A α -spectrum for 1 2 α 1 . Furthermore, we also show that the join K m ∨ P n ( m , n ≥ 2 ) is determined by its A α -spectrum for 1 2 α 1 . In the end, we pose some related open problems for future study.

Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d , r such that ( diag ( d ) − A ) r = 0 , where A is the adjacency matrix of G . We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices ( diag ( d ) − A ) ). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients 2 n − 1 n − 1 , and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.

It is well known that spectral Tur\'{a}n type problem is one of the most classical {problems} in graph theory. In this paper, we consider the spectral Tur\'{a}n type problem. Let $G$ be a graph and let $\mathcal{G}$ be a set of graphs, we say $G$ is \textit{$\mathcal{G}$-free} if $G$ does not contain any element of $\mathcal{G}$ as a subgraph. Denote by $\lambda_1$ and $\lambda_2$ the largest and the second largest eigenvalues of the adjacency matrix $A(G)$ of $G,$ respectively. In this paper we focus on the characterization of graphs without short odd cycles according to the adjacency eigenvalues of the graphs. Firstly, an upper bound on $\lambda_1^{2k}+\lambda_2^{2k}$ of $n$-vertex $\{C_3,C_5,\ldots,C_{2k+1}\}$-free graphs is established, where $k$ is a positive integer. All the corresponding extremal graphs are identified. Secondly, a sufficient condition for non-bipartite graphs containing an odd cycle of length at most $2k+1$ in terms of its spectral radius is given. At last, we characterize the unique graph having the maximum spectral radius among the set of $n$-vertex non-bipartite graphs with odd girth at least $2k+3,$ which solves an open problem proposed by Lin, Ning and Wu [Eigenvalues and triangles in graphs, Combin. Probab. Comput. 30 (2) (2021) 258-270]., Comment: 15 pages. It is accepted by Discrete Mathematics

The eccentricity matrix e ( G ) of a graph G is constructed from the distance matrix of G by keeping only the largest distances for each row and each column. This matrix can be interpreted as the opposite of the adjacency matrix obtained from the distance matrix by keeping only the distances equal to 1 for each row and each column. The e-eigenvalues of a graph G are those of its eccentricity matrix e ( G ) . Wang et al. [24] proposed the problem of determining the maximum e-spectral radius of trees with given order. In this paper, we consider the above problem of n-vertex trees with given diameter. The maximum e-spectral radius of n-vertex trees with fixed odd diameter is obtained, and the corresponding extremal trees are also determined. Recently, Wei et al. [22] determined all connected graphs on n vertices of maximum degree less than n − 1 , whose least eccentricity eigenvalues are in [ − 2 2 , − 2 ] . Denote by S n the star on n vertices. For tree T with order n ≥ 3 , it [22] was proved that e n ( T ) ≤ − 2 with equality if and only if T ≅ S n . According to the above results, the trees of order n ≥ 3 with least e-eigenvalues in [ − 2 2 , 0 ) are only S n . Motivated by [22] , we determine the trees with least e-eigenvalues in [ − 2 − 13 , − 2 2 ) .

In this paper, we study 24 possibilities of directed strongly regular graphs with adjacency matrix of rank 6, our proof is based on two effective algorithms. At last, we get 4 families of directed strongly regular graphs with realizable parameter sets, exclude 20 families of directed strongly regular graphs with feasible parameter sets; partially solve the classification problem of directed strongly regular graphs with adjacency matrix of rank 6.

The transition matrix of a graph G corresponding to the adjacency matrix A is defined by H ( t ) ≔ exp − i t A , where t ∈ R . The graph is said to exhibit pretty good state transfer between a pair of vertices u and v if there exists a sequence t k of real numbers such that lim k → ∞ H ( t k ) e u = γ e v , where γ is a complex number of unit modulus. We present a class of circulant graphs admitting pretty good state transfer. Also we find some circulant graphs not exhibiting pretty good state transfer. This generalizes several pre-existing results on circulant graphs admitting pretty good state transfer.

Graph energy is the sum of absolute values of the eigenvalues of the adjacency matrix of a graph. Eccentricity matrix, another graph matrix, was originally proposed, as D M A X matrix, by Randic (2013) [17] and redefined by Wang et al. (2018) [20] , by using the concept of the eccentricities of vertices. In this paper, we study the irreducibility and the spectrum of the eccentricity matrix for particular classes of graphs, namely the windmill graphs, the coalescence of complete graphs and the coalescence of two cycles. We further estimate the eccentricity energy and inertia for the graphs of these classes.

The index of a signed graph is the largest eigenvalue of its adjacency matrix. For positive integers n and m ≤ n 2 / 4 , we determine the maximum index of complete signed graphs with n vertices and m negative edges and characterize the signed graphs achieving this maximum. This settles (the corrected version of) a conjecture by Koledin and Stanic (2017).

Let Γ denote a bipartite distance-regular graph with diameter D ≥ 4 and valency k ≥ 3 . Let X denote the vertex set of Γ , and for any integer i , let Γ i ( x ) denote the set of vertices at distance i from x . Let V = ℂ X denote the vector space over ℂ consisting of column vectors whose coordinates are indexed by X and whose entries are in ℂ , and for z ∈ X let z denote the element of V with a 1 in the z coordinate and 0 in all other coordinates. Fix vertices x , u , v where u ∈ Γ 2 ( x ) and v ∈ Γ 2 ( x ) ∩ Γ 2 ( u ) , and let T = T ( x ) denote the Terwilliger algebra with respect to x . Under certain additional combinatorial assumptions, we give a combinatorially-defined spanning set for a T -module of endpoint 2,and we give the action of the adjacency matrix on this spanning set. The vectors in our spanning set are defined as sums and differences of vectors z , where the vertices z are chosen based on the their distances from x , u , and v . We use this T -module to construct combinatorially-defined bases for all isomorphism classes of irreducible T -modules of endpoint 2 for examples including the Doubled Odd graphs, the Double Hoffman–Singleton graph, Tutte’s 12-cage graph, and the Foster graph. We provide a list of several other graphs satisfying our conditions.

Let G be a graph with adjacency matrix A ( G ) and let D ( G ) be the diagonal matrix of the degrees of G . For every real α ∈ [ 0 , 1 ] , Nikiforov defined the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . In this paper, we study the k th largest eigenvalue of A α -matrix of graphs, where 1 ≤ k ≤ n . We present several upper and lower bounds on the k th largest eigenvalue of A α -matrix and characterize the extremal graphs corresponding to some of these obtained bounds. As applications, some bounds we obtained can generalize some known results on adjacency matrix and signless Laplacian matrix of graphs. Finally, we solve a problem proposed by Nikiforov (2017).

Let G be a simple undirected graph. The permanental sum of G is equal to the permanent of the matrix I + A ( G ) , where I is the identity matrix and A ( G ) is an adjacency matrix of G . The computation of permanental sum is #P-complete. In this paper, we compute the permanental sum of graphs of small sizes and derive explicit formulae for the permanental sum of graphs of large sizes. The results from small sizes are used as evidence to support a conjecture on an upper bound of the permanental sum of a graph in terms of its size. The results from large sizes are obtained by a new technique of employing rook’s polynomial via the Principle of Inclusion and Exclusion.

Let α ≠ β be two positive scalars. A Euclidean representation of a simple graph G in R r is a mapping of the nodes of G into points in R r such that the squared Euclidean distance between any two distinct points is α if the corresponding nodes are adjacent and β otherwise. A Euclidean representation is spherical if the points lie on an ( r − 1 ) -sphere, and is J -spherical if this sphere has radius 1 and α = 2 β . Let dim E ( G ) , dim S ( G ) and dim J ( G ) denote, respectively, the smallest dimension r for which graph G admits a Euclidean, spherical and J -spherical representation. In this paper, we extend and simplify the results of Roy (2010) and Nozaki and Shinohara (2012) by deriving exact simple formulas for dim E ( G ) and dim S ( G ) in terms of the eigenvalues of V T A V , where A is the adjacency matrix of G and V is the matrix whose columns form an orthonormal basis for the orthogonal complement of the vector of all 1’s. We also extend and simplify the results of Musin (2018) by deriving explicit formulas for determining the J -spherical representation of G and for determining dim J ( G ) in terms of the largest eigenvalue of A , the adjacency matrix of the complement graph G . As a by-product, we obtain several other related results and in particular we answer a question raised by Musin in Musin (2018).

Let G be a graph on n vertices and λ 1 , λ 2 , … , λ n its eigenvalues. The Estrada index of G is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. In this paper, we present some new lower bounds for the Estrada index of graphs and in particular of bipartite graphs that only depend on the number of vertices, the number of edges, Randic index, maximum and minimum degree and diameter.

The adjacency spectrum of a graph , which is denoted by Spec(), is the multiset of eigenvalues of its adjacency matrix. We say that two graphs and are cospectral if Spec()=Spec(). In this paper for each prime number p, p23, we construct a large family of cospectral non-isomorphic Cayley graphs over the dihedral group of order 2p.

An arithmetical structure on a finite, connected graph G is a pair of vectors ( d , r ) with positive integer entries for which ( d i a g ( d ) − A ( G ) ) r T = 0 , where d i a g ( d ) = d i a g ( d 1 , d 2 , … , d n ) , A ( G ) is the adjacency matrix of G and the entries of r have no common factor. In this paper, we will study the spectral radii of arithmetical structures on the path P n and determine the arithmetical structures with the minimal and maximal spectral radius on P n .

Let Γ be a simple undirected graph on a finite vertex set and let A be its adjacency matrix. Then Γ is singular if A is singular. The problem of characterizing singular graphs is easy to state but very difficult to resolve in any generality. In this paper we investigate the singularity of graphs for which the dihedral group acts transitively on vertices as a group of automorphisms.

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. A reduced graph G is said to be maximal if any reduced graph containing G as a proper induced subgraph has a higher rank. The main intent of this paper is to present some results on maximal graphs. First, we introduce a characterization of maximal trees (a reduced tree is a maximal tree if it is not a proper subtree of a reduced tree with the same rank). Next, we give a near-complete characterization of maximal ‘generalized friendship graphs.’ Finally, we present an enumeration of all maximal graphs with ranks 8 and 9. The ranks up to 7 were already done by Lepovic (1990), Ellingham (1993), and Lazic (2010).

Let G be a complex unit gain graph which is obtained from an undirected graph Γ by assigning a complex unit φ ( v i v j ) to each oriented edge v i v j such that φ ( v i v j ) φ ( v j v i ) = 1 for all edges. The Laplacian matrix of G is defined as L ( G ) = D ( G ) − A ( G ) , where D ( G ) is the degree diagonal matrix of Γ and A ( G ) = ( a i j ) has a i j = φ ( v i v j ) if v i is adjacent to v j and a i j = 0 otherwise. In this paper, we provide a combinatorial description of det ( L ( G ) ) that generalizes that for the determinant of the Laplacian matrix of a signed graph.

For a finite group G and an inverse closed subset S ⊆ G ∖ { e } , the Cayley graph X ( G , S ) has vertex set G and two vertices x , y ∈ G are adjacent if and only if x y − 1 ∈ S . Two graphs are called cospectral if their adjacency matrices have the same spectrum. Let p ≥ 3 be a prime number and T 4 p be the dicyclic group of order 4p. In this paper, with the help of the characters from representation theory, we construct a large family of pairwise non-isomorphic and cospectral Cayley graphs over the dicyclic group T 4 p with p ≥ 23 , and find several pairs of non-isomorphic and cospectral Cayley graphs for 5 ≤ p ≤ 19 .

Given a bipartite graph G = ( V , E ) with bipartition V = A ∪ B , where | A | = | B | . We present a new construction of pairs of cospectral bipartite graphs by “rotating” G in two different ways. More precisely, let l and k be two positive integers with l ≠ k . Take l copies of A (resp. B ) and k copies of B (resp. A ), say A 1 , A 2 , … , A l and B 1 , B 2 , … , B k (resp. B 1 ′ , B 2 ′ , … , B l ′ and A 1 ′ , A 2 ′ , … , A k ′ ). Let Γ (resp. Γ ′ ) be the graph with vertex set V ( Γ ) = A 1 ∪ A 2 ∪ ⋯ ∪ A l ∪ B 1 ∪ B 2 ∪ ⋯ ∪ B k (resp. V ( Γ ′ ) = B 1 ′ ∪ B 2 ′ ∪ ⋯ ∪ B l ′ ∪ A 1 ′ ∪ A 2 ′ ∪ ⋯ ∪ A k ′ ), such that the vertices of Γ (resp. Γ ′ ) are adjacent if and only if they are adjacent in G . We show that Γ and Γ ′ are cospectral with respect to the adjacency matrix, as well as the normalized Laplacian matrix. Moreover, we give a simple necessary and sufficient condition for Γ and Γ ′ to be non-isomorphic. The main ingredient of the proof involves a use of Hall’s Marriage Theorem.

An interval matrix is the adjacency matrix of an interval digraph or equivalently the biadjacency matrix of an interval bigraph. In this paper we investigate the forbidden substructures of an interval bigraph. Our method finds hitherto existing forbidden substructures for interval matrices, and via a more concise statement, as well as a new example showing that these substructures are not exhaustive.

A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. We prove that for a given nullity more than 1, there are only finitely many integral trees. It is also shown that integral trees with nullity 2 and 3 are unique., 14 pages, 4 figures; This is a through revision of the first version including the correction of Lemma 13 (of first version) which was not correct as stated. We thank a referee for pointing out this mistake

An odd [ 1 , b ] -factor of a graph G is a spanning subgraph H such that for each vertex v ∈ V ( G ) , d H ( v ) is odd and 1 ≤ d H ( v ) ≤ b . Let λ 3 ( G ) be the third largest eigenvalue of the adjacency matrix of G . For positive integers r ≥ 3 and even n , Lu et al. (2010) proved a lower bound for λ 3 ( G ) in an n -vertex r -regular graph G to guarantee the existence of an odd [ 1 , b ] -factor in G . In this paper, we improve the bound; it is sharp for every r .

The eccentricity matrix e ( G ) of a graph G is constructed from the distance matrix of G by keeping only the largest distances for each row and each column. This matrix can be interpreted as the opposite of the adjacency matrix obtained from the distance matrix by keeping only the distances equal to 1 for each row and each column. In this paper we focus on the eccentricity matrix of graphs. Let T be an n -vertex tree and let e n ( T ) be the least e -eigenvalue of T . On the one hand, we determine the n -vertex trees with the minimum e -spectral radius. On the other hand, for n ⩾ 3 , we show that e n ( T ) ⩽ − 2 with equality if and only if T is a star. As a consequence, we solve two conjectures proposed by Wang et al. (2018). Furthermore, we identify all the trees with given order and diameter having the minimum e -spectral radius . Finally, we determine all the n -vertex connected graphs whose maximum degrees are less than n − 1 and least e -eigenvalues are in [ − 2 2 , − 2 ] .

The work of Wang et al. (2020) established an upper bound on the multiplicity of a real number as an adjacency eigenvalue of an undirected simple graph G according to the dimension of its cycle space and the number of its pendants. The work of Cardoso et al. (2018) studied the multiplicity of α as an eigenvalue of α D ( G ) + ( 1 − α ) A ( G ) , α ∈ [ 0 , 1 ) , where D ( G ) is the diagonal matrix of degrees and A ( G ) is the adjacency matrix of G , which was ported to signed graphs by the work of Belardo et al. (2019). Here, on the one hand, we consider both the Wang–Wei–Jin-type and Cardoso–Pasten–Rojo-type routines developed for graphs to the level of mixed graphs and complex unit gain graphs. On the other hand, we consider Belardo–Brunetti–Ciampella-type routines developed for signed graphs to the level of mixed graphs and complex unit gain graphs. We show that, with suitable adaption, such routines can be successfully ported to mixed graphs and complex unit gain graphs. By these obtained results in the current paper, the corresponding results for undirected graphs, oriented graphs and signed graphs are deduced.