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HermitianRandić matrix and HermitianRandić energy of mixed graphs
Journal of Inequalities and Applications volume 2017, Article number: 54 (2017)
Abstract
Let M be a mixed graph and \(H(M)\) be its Hermitianadjacency matrix. If we add a Randić weight to every edge and arc in M, then we can get a new weighted Hermitianadjacency matrix. What are the properties of this new matrix? Motivated by this, we define the HermitianRandić matrix \(R_{H}(M)=(r_{h})_{kl}\) of a mixed graph M, where \((r_{h})_{kl}=(r_{h})_{lk}=\frac{\mathbf{i}}{\sqrt {d_{k}d_{l}}}\) (\(\mathbf{i}=\sqrt{1}\)) if \((v_{k},v_{l})\) is an arc of M, \((r_{h})_{kl}=(r_{h})_{lk}=\frac{1}{\sqrt{d_{k}d_{l}}}\) if \(v_{k}v_{l}\) is an undirected edge of M, and \((r_{h})_{kl}=0\) otherwise. In this paper, firstly, we compute the characteristic polynomial of the HermitianRandić matrix of a mixed graph. Furthermore, we give bounds on the HermitianRandić energy of a general mixed graph. Finally, we give some results about the HermitianRandić energy of mixed trees.
Introduction
In this paper, we only consider simple graphs without multiedges and loops. A graph M is said to be mixed if it is obtained from an undirected graph \(M_{U}\) by orienting a subset of its edges. We call \(M_{U}\) the underlying graph of M. Clearly, a mixed graph concludes both possibilities of all edges oriented and all edges undirected as extreme cases.
Let M be a mixed graph with vertex set \(V(M)=\{v_{1},v_{2},\ldots ,v_{n}\}\) and edge set \(E(M)\). For \(v_{i},v_{j}\in V(M)\), we denote an undirected edge joining two vertices \(v_{i}\) and \(v_{j}\) of M by \(v_{i}v_{j}\) (or \(v_{i}\leftrightarrow v_{j}\)). Denote a directed edge (or arc) from \(v_{i}\) to \(v_{j}\) by \((v_{i},v_{j})\) (or \(v_{i}\rightarrow v_{j}\)). In addition, let \(E_{0}(M)\) denote the set of all undirected edges and \(E_{1}(M)\) denote all the directed arcs set. Clearly, \(E(M)\) is the union of \(E_{0}(M)\) and \(E_{1}(M)\). A mixed graph is called mixed tree (or mixed bipartite graph) if its underlying graph is a tree (or bipartite graph). In general, the order, size, number of components and degree of a vertex of M are the same to those in \(M_{U}\). We use Bondy and Murty [1] for terminologies and notations not defined here.
Let G be a simple graph with vertex set \(\{v_{1},v_{2},\ldots,v_{n}\} \). The adjacency matrix of a simple graph G of order n is defined as the \(n\times n\) symmetric square matrix \(A=A(G)=(a_{ij})\), where \(a_{ij}=1\) if \(v_{i}v_{j}\) is an edge of G, otherwise \(a_{ij}=0\). We denote by \(d_{i}=d(v_{i})=d_{G}(v_{i})\) (\(i=1,2,\ldots,n\)) the degree of vertex \(v_{i}\). In addition, for a mixed graph M, if \(v_{i}\in V(M)\), then we also denote \(d_{i}=d(v_{i})=d_{M_{U}}(v_{i})\). The energy of the graph G (see the survey of Gutman, Li and Zhang [2] and the book of Li, Shi and Gutman [3]) is defined as \(\mathcal{E}_{A}(G)=\sum_{i=1}^{n}\rho_{i}\), where \(\rho_{1},\rho _{2},\ldots,\rho_{n}\) are all eigenvalues of \(A(G)\).
A convenient parameter of G is the general Randić index \(R_{\alpha}(G)\) defined as \(R_{\alpha}(G)=\sum_{uv\in E(G)}(d_{u}d_{v})^{\alpha}\), where the summation is over all (unordered) edges uv in G. The molecular structuredescriptor, first proposed by Randić [4] in 1975, is defined as the sum of \(\frac{1}{\sqrt{d_{u}d_{v}}}\) over all edges uv of G (with \(\alpha=\frac{1}{2}\)). Nowadays, \(R=R(G)=\sum_{uv\in E(G)}\frac{1}{\sqrt{d_{u}d_{v}}}\) of G is referred to as the Randić index. Countless chemical applications, the mathematical properties and mathematical chemistry of the Randić index were reported in [5–7].
Gutman et al. [8] pointed out that the Randićindexconcept is purposeful to associate the graph G with a symmetric square matrix of order n, named Randić matrix \(R(G)=(r_{ij})\), where \(r_{ij}=\frac{1}{\sqrt{d_{i}d_{j}}}\) if \(v_{i}v_{j}\) is an edge of G, otherwise \(r_{ij}=0\). Let \(D(G)\) be the diagonal matrix of vertex degrees of G. If G has no isolated vertices, then \(R(G)=D(G)^{\frac {1}{2}}A(G)D(G)^{\frac{1}{2}}\).
The concept of Randić energy of a graph G, denoted by \(\mathcal{E}_{R}(G)\), was introduced in [9] as \(\mathcal {E}_{R}(G)=\sum_{i=1}^{n}\gamma_{i}\), where \(\gamma_{i}\) is the eigenvalues of \(R(G)\), \(i=1,2,\ldots,n\). Some basic properties of the Randić index, Randić matrix and Randić energy were determined in the papers [8–20].
An oriented graph \(G^{\sigma}\) is a digraph which assigns each edge of G a direction σ. The skew adjacency matrix associated to \(G^{\sigma}\) is the \(n\times n\) matrix \(S(G^{\sigma})=(s_{ij})\), where \(s_{ij}=s_{ji}=1\) if \((v_{i},v_{j})\) is an arc of \(G^{\sigma}\), otherwise \(s_{ij}=s_{ji}=0\). The skew energy of \(G^{\sigma}\), denoted by \(\mathcal{E}_{S}(G^{\sigma})\), is defined as the sum of the norms of all the eigenvalues of \(S(G^{\sigma})\). For more details about skew energy, we can refer to [21, 22].
In 2016, Gu, Huang and Li [14] defined the skew Randić matrix \(R_{s}(G^{\sigma})=((r_{s})_{ij})\) of an oriented graph \(G^{\sigma}\) of order n, where \((r_{s})_{ij}=(r_{s})_{ji}=\frac {1}{\sqrt{d_{i}d_{j}}}\) if \((v_{i},v_{j})\) is an arc of \(G^{\sigma}\), otherwise \((r_{s})_{ij}=(r_{s})_{ji}=0\). Let \(D(G)\) be the diagonal matrix of vertex degrees of G. If \(G^{\sigma}\) has no isolated vertices, then \(R_{s}(G^{\sigma})=D(G)^{\frac{1}{2}}S(G^{\sigma })D(G)^{\frac{1}{2}}\).
The Hermitianadjacency matrix of a mixed graph M of order n is the \(n\times n\) matrix \(H(M)=(h_{kl})\), where \(h_{kl}=h_{lk}= \mathbf{i}\) (\(\mathbf{i}=\sqrt{1}\)) if \((v_{k},v_{l})\) is an arc of M, \(h_{kl}=h_{lk}=1\) if \(v_{k}v_{l}\) is an undirected edge of M, and \(h_{kl}=0\) otherwise. Obviously, \(H(M)=H(M)^{\ast}:={\overline{H(M)}}^{T}\). Thus all its eigenvalues are real. This matrix was introduced by Liu and Li in [23] and independently by Guo and Mohar in [24]. The Hermitian energy of a mixed graph M is defined as \(\mathcal {E}_{H}(M)=\sum^{n}_{i=1}\lambda_{i}\), where \(\lambda_{1},\lambda _{2},\ldots,\lambda_{n}\) are all eigenvalues of \(H(M)\). Denote by \(\operatorname{Sp}_{H}(M)=(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\) the spectrum of \(H(M)\). For more details about the Hermitianadjacency matrix and the Hermitian energy of mixed graphs, we can refer to [23–28].
From the above we can see that if we add a Randić weight to every edge in a simple graph G, then we can get a Randić matrix \(R(G)\). If we add a Randić weight to every arc in an oriented graph \(G^{\sigma}\), then we can get a skew Randić matrix \(R_{s}(G^{\sigma})\). Let M be a mixed graph and \(H(M)\) be its Hermitianadjacency matrix. If we add a Randić weight to every edge and arc in M, then we can get a new weighted Hermitianadjacency matrix. What are the properties of this new matrix? Motivated by this, we define the HermitianRandić matrix of a mixed graph M.
Let M be a mixed graph on the vertex set \(\{v_{1},v_{2},\ldots,v_{n}\} \), then the HermitianRandić matrix of M is the \(n\times n\) matrix \(R_{H}(M)=((r_{h})_{kl})\), where
Let \(D(M_{U})\) be the diagonal matrix of vertex degrees of \(M_{U}\). If M has no isolated vertices, then \(R_{H}(M)=D(M_{U})^{\frac {1}{2}}H(M)D(M_{U})^{\frac{1}{2}}\). For a mixed graph M, let \(R_{H}(M)\) be its HermitianRandić matrix. It is obvious that \(R_{H}(M)\) is a Hermitian matrix, so all its eigenvalues \(\mu_{1},\mu_{2},\ldots,\mu_{n}\) are real. The spectrum of \(R_{H}(M)\) is defined as \(\operatorname{Sp}_{R_{H}}(M)=(\mu_{1},\mu _{2},\ldots,\mu_{n})\). The energy of \(R_{H}(M)\), denoted by \(\mathcal {E}_{R_{H}}(M)\), is called HermitianRandić energy, which is defined as the sum of the absolute values of its eigenvalues of \(R_{H}(M)\), that is, \(\mathcal{E}_{R_{H}}(M)=\sum^{n}_{i=1}\mu_{i}\).
In this paper, we define the HermitianRandić matrix of a mixed graph M and study some basic characteristics of the HermitianRandić matrix of mixed graphs. In Section 2, we give the characteristic polynomial of the HermitianRandić matrix of a mixed graph M. In Section 3, we study some bounds on the HermitianRandić energy of mixed graphs with different parameters and give the conditions under which mixed graphs can attain those HermitianRandić energy bounds. In Section 4, we show that the HermitianRandić energy of a mixed tree is the same as the Randić energy of its underlying graph. In Section 5, we summarize the results of this paper and give some future works we will study.
HermitianRandić characteristic polynomial of a mixed graph
In this section, we will give the characteristic polynomial of a HermitianRandić matrix of a mixed graph M, i.e., the \(R_{H}\)characteristic polynomial of M. At first, we will introduce some basic definitions.
The value of a mixed walk \(W=v_{1}v_{2}\cdots v_{l}\) is \(r_{h}(W)=(r_{h})_{12}(r_{h})_{23}(r_{h})_{(l1)l}\). A mixed walk W is positive (or negative) if \(r_{h}(W)=\frac{1}{\sqrt {d_{1}d_{l}}d_{2}d_{3}\cdots d_{(l1)}}\) (or \(r_{h}(W)=\frac{1}{\sqrt {d_{1}d_{l}}d_{2}d_{3}\cdots d_{(l1)}}\)). Note that for one direction the value of a mixed walk or a mixed cycle is α, then for the reversed direction its value is α̅. Thus, if the value of a mixed cycle C is \(\prod_{v_{j}\in V(C)}\frac{1}{d(v_{j})}\) (resp. \(\prod_{v_{j}\in V(C)}\frac{1}{d(v_{j})}\)) in a direction, then its value is \(\prod_{v_{j}\in V(C)}\frac{1}{d(v_{j})}\) (resp. \(\prod_{v_{j}\in V(C)}\frac{1}{d(v_{j})}\)) for the reversed direction. In these situations, we just term this mixed cycle a positive (resp. negative) mixed cycle without mentioning any direction.
If each mixed cycle is positive (resp. negative) in a mixed graph M, then M is positive (resp. negative). A mixed graph M is called an elementary graph if every component of M is an edge, an arc or a mixed cycle, where every edgecomponent in M is defined to be positive. A real spanning elementary subgraph of a mixed graph M is an elementary subgraph such that it contains all vertices of M and all its mixed cycles are real.
Now we will give two results which are similar to those in [23, 29, 30].
Let M be a mixed graph of order n with its HermitianRandić matrix \(R_{H}(M)\). Denote the \(R_{H}\) characteristic polynomial of \(R_{H}(M)\) of M by
Theorem 2.1
Let \(R_{H}(M)\) be the HermitianRandić matrix of a mixed graph M of order n. Then
where the summation is over all real spanning elementary subgraphs \(M'\) of M, \(r(M')=nc(M')\), \(c(M')\) denotes the number of components of \(M'\), \(l(M')\) denotes the number of negative mixed cycles of \(M'\), \(s(M')\) denotes the number of mixed cycles with length ≥3 in \(M'\), \(W(M')=\prod_{v_{i}\in V(M')}\frac{1}{d_{M_{U}}(v_{i})}\).
Proof
Let M be a mixed graph of order n with vertex set \(\{ v_{1},v_{2},\ldots, v_{n}\}\). Then
where \(S_{n}\) is the set of all permutations on \(\{1,2,\ldots,n\}\).
Consider a term \(\operatorname{sgn}(\pi)(r_{h})_{1\pi(1)}(r_{h})_{2\pi(2)}\cdots (r_{h})_{n\pi(n)}\) in the expansion of \(\operatorname{det}R_{H}(M)\). If \(v_{k}v_{\pi(k)}\) is not an edge or arc of M, then \((r_{h})_{k\pi (k)}=0\); that is, this term vanishes. Thus, if the term corresponding to a permutation π is nonzero, then π is fixedpointfree and can be expressed uniquely as the composition of disjoint cycles of length at least 2. Consequently, each nonvanishing term in the expansion of \(\operatorname{det}R_{H}(M)\) gives rise to an elementary mixed graph \(M'\) of M with \(V(M')=V(M)\). That is, \(M'\) is a spanning elementary subgraph of M of order n.
A spanning elementary subgraph \(M'\) of M with \(s(M')\) number of mixed cycles (length ≥3) gives \(2^{s(M')}\) permutations π since, for each mixed cyclecomponent in \(M'\), there are two ways of choosing the corresponding cycles in π. For a vertex \(v_{k}\in V(M')\), we denote \(d_{k}=d(v_{k})=d_{M_{U}}(v_{k})\). Furthermore, if for some direction of a permutation π, a mixed cyclecomponent \(C_{1}\) has value \(\mathbf{i}\prod_{v_{j}\in V(C_{1})}\frac{1}{d(v_{j})}\) (or \(\mathbf{i}\prod_{v_{j}\in V(C_{1})}\frac{1}{d(v_{j})}\)), then for the other direction \(C_{1}\) has value \(\mathbf {i}\prod_{v_{j}\in V(C_{1})}\frac{1}{d(v_{j})}\) (or \(\mathbf {i}\prod_{v_{j}\in V(C_{1})}\frac{1}{d(v_{j})}\)) and vice versa. Thus, they cancel each other in the summation. In addition, if for some direction of a permutation π, \(C_{1}\) has value \(\prod_{v_{j}\in V(C_{1})}\frac{1}{d(v_{j})}\) (or \(\prod_{v_{j}\in V(C_{1})}\frac {1}{d(v_{j})}\)), then for the other direction \(C_{1}\) has the same value. For each edgecomponent \((kl)\) corresponding to the factors \((r_{h})_{kl}(r_{h})_{lk}\) has value \(\frac{1}{\sqrt{d_{k}d_{l}}}\frac {1}{\sqrt{d_{l}d_{k}}}=\frac{1}{d_{k}d_{l}}\). For each arccomponent \((kl)\) corresponding to the factors \((r_{h})_{kl}(r_{h})_{lk}\) has value \(\frac{\mathbf{i}\cdot(\mathbf{i})}{\sqrt {d_{k}d_{l}}\sqrt{d_{l}d_{k}}}=\frac{1}{d_{k}d_{l}}\).
Since \(\operatorname{sgn}(\pi)=(1)^{nc(M')}=(1)^{r(M')}\) and each real spanning elementary subgraph \(M'\) contributes \((1)^{r(M')+l(M')}2^{s(M')}\prod_{v_{i}\in V(M')}\frac{1}{d_{M_{U}}(v_{i})}\) to the determinant of \(R_{H}(M)\). This completes the proof. □
Now, we shall obtain a description of all the coefficients of the characteristic polynomial \(P_{R_{H}}(M,x)\) of a mixed graph M.
Theorem 2.2
For a mixed graph M, if the \(R_{H}\)characteristic polynomial of M is denoted by \(P_{R_{H}}(M,x)=\operatorname {det}(xIR_{H}(M))=x^{n}+a_{1}x^{n1}+a_{2}x^{n2}+\cdots+a_{n}\), then the coefficients of \(P_{R_{H}}(M,x)\) are given by
where the summation is over all real elementary subgraphs \(M'\) with order k of M, \(r(M')=kc(M')\), \(c(M')\) denotes the number of components of \(M'\), \(l(M')\) denotes the number of negative mixed cycles of \(M'\), \(s(M')\) denotes the number of mixed cycles with length ≥3 in \(M'\).
Proof
The proof follows from Theorem 2.1 and the fact that \((1)^{k}a_{k}\) is the summation of determinants of all principal \(k\times k\) submatrices of \(R_{H}(M)\). □
Corollary 2.3
For a mixed graph M, let the \(R_{H}\)characteristic polynomial of M be denoted by \(P_{R_{H}}(M,x)=\operatorname {det}(xIR_{H}(M))=x^{n}+a_{1}x^{n1}+a_{2}x^{n2}+\cdots+a_{n}\).

(1)
If M is a mixed tree, then \((1)^{k}a_{k}=\sum_{M'}(1)^{r(M')}\prod_{v_{i}\in V(M')}\frac{1}{d_{M_{U}}(v_{i})}\).

(2)
If M is a mixed graph and its underlying graph \(M_{U}\) is r regular (\(r\neq0\)), then \((1)^{k}a_{k}=\sum_{M'}(1)^{r(M')+l(M')}2^{s(M')}\frac{1}{r^{k}}\).

(3)
If M is a mixed bipartite graph, then all coefficients of \(a_{\mathrm{odd}}\) are equal to 0, and its spectrum is symmetry about 0.
Note that if M is a positive mixed graph, then for every real elementary subgraph \(M'\) of M, we have
Then \(P_{R_{H}}(M,x)=P_{R_{H}}(M_{U},x)\), that is to say,
Theorem 2.4
If M is a positive mixed graph and \(M_{U}\) be its underlying graph, then \(\operatorname{Sp}_{R_{H}}(M)=\operatorname{Sp}_{R_{H}}(M_{U})\).
Bounds on the HermitianRandić energy of mixed graphs
In this section, we will give some bounds on the HermitianRandić energy of mixed graphs. First, we will give some properties of a HermitianRandić matrix of mixed graphs.
Lemma 3.1
Let M be a mixed graph of order \(n\geq1\).

(1)
\(\mathcal{E}_{R_{H}}(M)=0\) if and only if \(M\cong\overline{K}_{n}\).

(2)
If \(M=M_{1}\cup M_{2}\cup\cdots\cup M_{p}\), then \(\mathcal {E}_{R_{H}}(M)=\mathcal{E}_{R_{H}}(M_{1})+\mathcal {E}_{R_{H}}(M_{2})+\cdots+\mathcal{E}_{R_{H}}(M_{p})\).
From Lemma 3.1, we can obtain the following theorem.
Theorem 3.2
Let M be a mixed graph with vertex set \(V(M)=\{v_{1},v_{2},\ldots ,v_{n}\}\), and \(d_{k}\) is the degree of \(v_{k}\), \(k=1,2,\ldots,n\). Let \(H(M)\) and \(R_{H}(M)\) be the Hermitianadjacency matrix and the HermitianRandić matrix of M, respectively. If M has isolated vertices, then \(\operatorname{det}H(M)=\operatorname{det}R_{H}(M)=0\). If M has no isolated vertices, then
Proof
If M has l isolated vertices, then \(M=M'\cup\overline {K}_{l}\), where \(M'\) has no isolated vertices. By Lemma 3.1, we have \(\operatorname{Sp}_{R_{H}}(M)=\operatorname{Sp}_{R_{H}}(M')\cup\{0, l\text{ times}\}\) and an analogous relation holds for Hermitianadjacency spectrum of M. That is, \(H(M)\) and \(R_{H}(M)\) have zero eigenvalues, therefore their determinants are equal to zero.
If M has no isolated vertices, then \(R_{H}(M)=D(M_{U})^{\frac {1}{2}}H(M)D(M_{U})^{\frac{1}{2}}\) is applicable, where \(D(M_{U})\) is the diagonal matrix of vertex degrees. The matrices \(R_{H}(M)\) and \(D(M_{U})^{\frac{1}{2}}R_{H}(M)D(M_{U})^{\frac{1}{2}}\) are similar and thus have equal eigenvalues. We have
therefore,
So,
This completes the proof. □
Similar to Theorem 3.2, we can obtain the following theorem.
Theorem 3.3
If M is a mixed graph with vertex set \(V(M)=\{v_{1},v_{2},\ldots ,v_{n}\}\) and its underlying graph \(M_{U}\) is r regular, then \(\mathcal{E}_{R_{H}}(M)=\frac{1}{r}\mathcal{E}_{H}(M)\). In addition, if \(r=0\), then \(\mathcal{E}_{R_{H}}(M)=0\).
Proof
If \(r=0\), then M is the graph that has no edges. Then all the entries of \(R_{H}(M)\) are equal to 0, i.e., \(R_{H}(M)=\mathbf {0}\). Similarly, \(H(M)=\mathbf{0}\). Since all eigenvalues of the zero matrix are equal to 0, hence \(\mathcal{E}_{R_{H}}(M)=\mathcal {E}_{H}(M)=0\).
If \(r>0\), i.e., M is regular of degree \(r>0\), then \(d_{1}=d_{2}=\cdots =d_{n}=r\), where \(d_{k}\) is the degree of \(v_{k}\), \(k=1,2,\ldots,n\). Hence, \((r_{h})_{sk}=(r_{h})_{ks}=\frac{\mathbf{i}}{r}\) if \((v_{s},v_{k})\) is an arc of M, \((r_{h})_{sk}=(r_{h})_{ks}=\frac {1}{r}\) if \(v_{s}v_{k}\) is an undirected edge of M, and \((r_{h})_{sk}=0\) otherwise.
This implies that \(R_{H}(M)=\frac{1}{r}H(M)\). Therefore, \(\mu _{i}=\frac{1}{r}\lambda_{i}\), where \(\mu_{i}\) is the eigenvalue of \(R_{H}(M)\), and \(\lambda_{i}\) is the eigenvalue of \(H(M)\) for \(i=1,2,\ldots,n\). Then this theorem follows from the definitions of \(\mathcal{E}_{R_{H}}(M)\) and \(\mathcal{E}_{H}(M)\). □
Similar to the results about the skew Randić energy in [14], we can establish the following lower and upper bounds for the HermitianRandić energy. First, we need the following theorem. Here and later, \(\mathbf{I}_{n}\) denotes the unit matrix of order n.
Theorem 3.4
Let M be a mixed graph of order n and \(\mu_{1}\geq\mu_{2}\geq\cdots \geq\mu_{n}\) be the HermitianRandić spectrum of \(R_{H}(M)\). Then \(\mu_{1}=\mu_{2}=\cdots=\mu_{n}\) if and only if there exists a constant \(c=\mu_{i}^{2}\) for all i such that \(R^{2}_{H}(M)=c \mathbf{I}_{n}\).
Proof
Let \(\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{n}\) be the HermitianRandić spectrum of \(R_{H}(M)\). Then there exists a unitary matrix U such that
So,
where c is a constant and \(c=\mu_{i}^{2}\) for all i.
This completes the proof. □
Theorem 3.5
Let M be a mixed graph of order n and \(\mu_{1}\geq\mu_{2}\geq\cdots \geq\mu_{n}\) be the HermitianRandić spectrum of \(R_{H}(M)\). Let \(M_{U}\) be the underlying graph of M, \(p=\operatorname{det}R_{H}(M)\). Then
with equalities holding both in the lower bound and upper bound if and only if there exists a constant \(c=\mu_{i}^{2}\) for all i such that \(R^{2}_{H}(M)=c \mathbf{I}_{n}\).
Proof
Let \(\{\mu_{1},\mu_{2},\ldots,\mu_{n}\}\) be the HermitianRandić spectrum of M, where \(\mu_{1}\geq\mu_{2}\geq \cdots\geq\mu_{n}\). Since \(\sum_{j=1}^{n}\mu_{j}^{2}=\operatorname{tr}(R_{H}^{2}(M))=\sum_{j=1}^{n}\sum_{k=1}^{n}(r_{h})_{jk}(r_{h})_{kj}=\sum_{j=1}^{n}\sum_{k=1}^{n}(r_{h})_{jk}\overline{(r_{h})}_{jk} =\sum_{j=1}^{n}\sum_{k=1}^{n}(r_{h})_{jk}^{2}=2R_{1}(M_{U})\), where \(R_{1}(M_{U})=\sum_{v_{j}v_{k}\in E(M_{U})}\frac{1}{d_{j}d_{k}}\) (unordered).
Applying the CauchySchwarz inequality, we have
On the other hand,
By using an arithmetic geometric average inequality, we can get that
Therefore, we can obtain the lower bound on the HermitianRandić energy
From the CauchySchwarz inequality and the arithmetic geometric average inequality, we know that the equalities hold both in the lower bound and upper bound if and only if \(\mu_{1}=\mu_{2}=\cdots=\mu _{n}\), i.e., there exists a constant \(c=\mu_{i}^{2}\) for all i such that \(R^{2}_{H}(M)=c \mathbf{I}_{n}\).
This completes the proof. □
Corollary 3.6
Let M be a mixed graph and its underlying graph \(M_{U}\) be r (≠0) regular and \(E(M_{U})=m\). Let \(\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu _{n}\) be the HermitianRandić spectrum of \(R_{H}(M)\). Then
where \(p=\operatorname{det}R_{H}(M)\), with equalities holding both in the lower bound and upper bound if and only if \(\frac{1}{r}=\mu_{i}^{2}\) for all i such that \(R^{2}_{H}(M)=\frac{1}{r} \mathbf{I}_{n}\).
Proof
If M is a mixed graph and its underlying graph \(M_{U}\) is r regular, then \(R_{1}(M_{U})=\frac{m}{r^{2}}\) and \(2m=nr\). By Theorems 3.4 and 3.5, we can obtain the results. □
Lemma 3.7
[19]
Let G be a graph of order n with no isolated vertices. Then
with equality in the lower bound if and only if G is a complete graph, and equality in the upper bound if and only if either

(1)
n is even and G is the disjoint union of \(n/2\) paths of length 1, or

(2)
n is odd and G is the disjoint union of \((n3)/2\) paths of length 1 and one path of length 2.
Combining Theorem 3.5 and Lemma 3.7, we can get upper and lower bounds for the HermitianRandić energy by replacing \(R_{1}(M_{U})\) with other parameters. We now give bounds of the HermitianRandić energy of a mixed graph with respect to its order.
Theorem 3.8
Let M be a mixed graph of order \(n\geq3\) without isolated vertices and \(M_{U}\) be its underlying graph. Let \(\mu_{1}\geq\mu_{2}\geq\cdots \geq\mu_{n}\) be the HermitianRandić spectrum of \(R_{H}(M)\). Then
The equality in the upper bound holds if and only if n is even and \(M_{U}\) is the disjoint union of \(n/2\) paths of length 1. The equality in the lower bound holds if and only if \(M_{U}\) is a complete graph and \(\mu_{1}=\mu_{n}\neq0\), \(\mu_{j}=0\), \(j=2,\ldots,n1\).
Proof
Let \(R_{H}(M)\) be the HermitianRandić matrix of M and \(\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{n}\) be the HermitianRandić spectrum of \(R_{H}(M)\).
For the upper bound, combining Lemma 3.7 and \(\mathcal {E}_{R_{H}}(M)\leq\sqrt{2nR_{1}(M_{U})}\) of Theorem 3.5, we have
From Theorem 3.5 and Lemma 3.7, we know that the equality in the upper bound holds if and only if n is even, \(M_{U}\) is the graph described in Lemma 3.7(1), and \(\mu_{1}=\mu_{2}=\cdots=\mu_{n}\), that is, we can obtain the upper bound when n is even and \(M_{U}\) is the disjoint union of \(n/2\) paths of length 1.
For the lower bound, since the sum of the diagonal entries of \(R_{H}(M)\) is 0, i.e., \(\sum^{n}_{k=1}\mu_{k}=0\), then
Hence, \(\sum_{k< l}\mu_{k}\mu_{l}=R_{1}(M_{U})\).
From the definition of the HermitianRandić energy of a mixed graph, we have
Combining this with Lemma 3.7, we have
So,
From the proof above and Lemma 3.7, we know that the equality in the lower bound holds if and only if \(M_{U}\) is a complete graph and \(\mu_{k}\mu_{l}\geq0\) or \(\mu_{k}\mu _{l}\leq0\) for all \(1\leq k< l\leq n\). Note that \(\sum_{k=1}^{n}\mu _{k}=0\) and M has no isolated vertices, so the former case can not happen. Hence, the equality in the lower bound holds if and only if \(M_{U}\) is a complete graph and \(\mu_{1}=\mu_{n}\neq0\), \(\mu_{j}=0\), \(j=2,\ldots,n1\).
This completes the proof. □
Remark 3.9
It should be pointed out that when M is a complete mixed graph, its HermitianRandić spectrum is not unique. For example, let \(M_{U}=K_{3}\), if all edges of \(E(M)\) are oriented, then we have \(\mu _{1}=\mu_{3}=\frac{\sqrt{3}}{2}\), \(\mu_{2}=0\), then we can obtain the lower bound in Theorem 3.8. If some edges of \(E(M)\) are undirected, then we can not obtain the lower bound in Theorem 3.8. For example, if \((r_{h})_{12}=(r_{h})_{32}=\frac{i}{2}\), \((r_{h})_{13}=\frac{1}{2}\), then \(\mu_{1}=1\) and \(\mu_{2}=\mu_{3}=\frac {1}{2}\). Hence, the problem of determining all complete mixed graphs for which the lower bound in Theorem 3.8 is attained appears to be somewhat more difficult.
To deduce more bounds on \(\mathcal{E}_{R_{H}}(M)\), the following lemma is needed.
Lemma 3.10
[31]
Let \(\mathbf{x}, \mathbf{y}\in\mathbb{R}^{n}\) and let \(A(\mathbf {x})=\frac{1}{n}\sum^{n}_{i=1}x_{i}\), \(A(\mathbf{y})=\frac{1}{n}\sum^{n}_{i=1}y_{i}\). If \(\phi\leq x_{i}\leq\Phi\) and \(\gamma\leq y_{i}\leq \Gamma\), then
Now we turn to new bounds on \(\mathcal{E}_{R_{H}}(M)\).
Theorem 3.11
Let M be a mixed graph of order n and \(M_{U}\) be its underlying graph. Let \(\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{n}\) be the HermitianRandić spectrum of \(R_{H}(M)\). Then
where \(\alpha=\min_{1\leq i\leq n}\{\mu_{i}\}\), \(\beta=\max\{\mu _{1},\mu_{n}\}\).
Proof
Note that
Let \(S=\sum_{1\leq i\neq j\leq n}\mu_{i}\mu_{j}\), \(x_{i}=\mu_{i}\) and \(y_{i}=\mathcal{E}_{R_{H}}(M)\mu_{i}\), \(i=1,2,\ldots,n\). Then \(S=\sum^{n}_{i=1}x_{i}y_{i}\).
From the definitions of α and β, we have \(\alpha\leq x_{i}\leq\beta\) and \(\mathcal{E}_{R_{H}}(M)\beta\leq y_{i}\leq{\mathcal {E}_{R_{H}}(M)\alpha}\). In addition, let \(A(\mathbf{x})=\frac{1}{n}\sum^{n}_{i=1}x_{i}=\frac{\mathcal{E}_{R_{H}}(M)}{n}\) and \(A(\mathbf {y})=\frac{1}{n}\sum^{n}_{i=1}y_{i}=\frac{(n1)\mathcal {E}_{R_{H}}(M)}{n}\). Hence, by Lemma 3.10, we have
It follows that
This together with (2) implies that
So,
This completes the proof. □
Note that the righthand side of (1) is a nondecreasing function on \(\alpha\geq0\). Combining this with Theorem 3.4, we have the following corollary.
Corollary 3.12
Let M be a mixed graph of order n and \(M_{U}\) be its underlying graph. Let \(\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{n}\) be the HermitianRandić spectrum of \(R_{H}(M)\). Then
where \(\beta=\max\{\mu_{1},\mu_{n}\}\). The equality holds if and only if \(R^{2}_{H}(M)=c \mathbf{I}_{n}\), where c is a constant such that \(\mu_{i}^{2}=c\) for all i.
In particular, if M is a connected mixed bipartite graph, then we have the following theorem.
Theorem 3.13
Let M be a connected mixed bipartite graph of order n and \(M_{U}\) be its underlying graph. Let \(\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{n}\) be the HermitianRandić spectrum of \(R_{H}(M)\). Then
where \(\alpha=\min_{1\leq i\leq\lfloor\frac{n}{2}\rfloor}\{\mu _{i}\}\).
Proof
Note that \(M_{U}\) is a bipartite graph. By Corollary 2.3(3), we have \(\mu_{i}=\mu_{n+1i}\) and \(\mu_{i}\geq0\) for \(i=1,2,\ldots,\lfloor\frac{n}{2}\rfloor\). Therefore,
Let \(T=\sum_{1\leq i\neq j\leq\lfloor\frac{n}{2}\rfloor}\mu_{i}\mu _{j}\), \(x_{i}=\mu_{i}\) and \(y_{i}=\frac{\mathcal{E}_{R_{H}}(M)}{2}\mu _{i}\), \(i=1,2,\ldots,\lfloor\frac{n}{2}\rfloor\). Then \(T=\sum^{\lfloor\frac{n}{2}\rfloor}_{i=1}x_{i}y_{i}\).
From the definition of α, we have \(\alpha\leq x_{i}\leq\mu_{1}\) and \(\frac{\mathcal{E}_{R_{H}}(M)}{2}\mu_{1}\leq y_{i}\leq\frac {\mathcal{E}_{R_{H}}(M)}{2}\alpha\). In addition, let \(A(\mathbf {x})=\frac{1}{\lfloor\frac{n}{2}\rfloor}\sum^{\lfloor\frac{n}{2}\rfloor }_{i=1}x_{i}=\frac{\mathcal{E}_{R_{H}}(M)}{2\lfloor\frac{n}{2}\rfloor}\) and \(A(\mathbf{y})=\frac{1}{\lfloor\frac{n}{2}\rfloor}\sum^{\lfloor \frac{n}{2}\rfloor}_{i=1}y_{i}=\frac{(\lfloor\frac{n}{2}\rfloor 1)\mathcal{E}_{R_{H}}(M)}{2\lfloor\frac{n}{2}\rfloor}\). Hence, by Lemma 3.10, we have
It follows that
This together with (4) implies that
So,
This completes the proof. □
Note that the righthand side of (3) is a nondecreasing function on \(\alpha\geq0\). Combining this with Theorem 3.4, we have the following corollary.
Corollary 3.14
Let M be a connected mixed bipartite graph of order n and \(M_{U}\) be its underlying graph. Let \(\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{n}\) be the HermitianRandić spectrum of \(R_{H}(M)\). Then
the equality holds if and only if \(R^{2}_{H}(M)=c \mathbf {I}_{n}\), where c is a constant such that \(\mu_{i}^{2}=c\) for all i.
HermitianRandić energy of trees
In [21], the authors proved that the skew energy of a directed tree is independent of its orientation. In [14], the authors showed that the skew Randić energy of a directed tree has the same result. In this section, we will show that the HermitianRandić energy also has the same result. In the beginning of this section, we first characterize the mixed graphs with cutedge.
Theorem 4.1
Let M be a mixed graph of order n, and \(e=uv\) is an edge of M. If uv is a cutedge of \(M_{U}\), where \(M_{U}\) is the underlying graph of M, then the spectrum and energy of \(R_{H}(M)\) are unchanged when the edge uv is replaced with a single arc uv or vu and vice versa.
Proof
Let \(e=uv\) be a cutedge of \(M_{U}\). Suppose that \(M_{1}\) is the graph obtained from M by replacing the edge uv with the arc uv or vu. Let \(M'\) and \(M_{1}'\) be real elementary subgraphs of order k of M and \(M_{1}\), respectively. If \(M'\) does not contain the cutedge uv, then \(M'\) is also a real elementary subgraph of \(M_{1}\), that is, \(M'=M_{1}'\). By Theorem 2.2, we have
If \(M'\) contains the cutedge uv, then there is a real elementary subgraph \(M_{1}'\) of \(M_{1}\) only different from \(M'\) on uv. Since uv is a cutedge of \(M_{U}\), uv is not contained in any cycles of \(M_{U}\). Hence, by Theorem 2.2, we have
Combining (5) and (6), we have \(a_{k}(M)a_{k}(M_{1})=0\) for any integer k.
Thus \(\operatorname{Sp}_{R_{H}}(M)=\operatorname{Sp}_{R_{H}}(M_{1})\). Moreover, \(\mathcal {E}_{{R_{H}}}(M)=\mathcal{E}_{{R_{H}}}(M_{1})\).
Similarly, we can prove that \(\operatorname{Sp}_{R_{H}}(M)=\operatorname{Sp}_{R_{H}}(M_{2})\) and \(\mathcal{E}_{{R_{H}}}(M)=\mathcal{E}_{{R_{H}}}(M_{2})\), where \(M_{2}\) is the mixed graph obtained from M by replacing the arc uv or vu with the edge uv. □
Thus, the HermitianRandić spectrum and the HermitianRandić energy are invariants when reversing the cutarc’s orientation or unorienting it or orienting an undirected cutedge. By applying Theorem 4.1, we can obtain the following corollaries.
Corollary 4.2
Let T be a mixed tree of order n and \(T'\) be the mixed tree obtained from T by reversing the orientations of all the arcs incident with a particular vertex of T. Then \(\mathcal{E}_{R_{H}}(T)=\mathcal{E}_{R_{H}}(T')\).
Corollary 4.3
Let T be a mixed tree and \(T_{U}\) be its underlying graph. Then

(1)
The HermitianRandić energy of T is independent of its orientation of the arc set.

(2)
The HermitianRandić energy of T is the same as the Randić energy of \(T_{U}\).
Conclusions
In this paper, we define the HermitianRandić matrix of a mixed graph M and give the definitions of HermitianRandić characteristic polynomial and HermitianRandić energy of a mixed graph M. We give the bounds on the HermitianRandić energy of a mixed graph M with respect to its order, the HermitianRandić spectrum and a general Randić index (with \(\alpha=1\)). We also obtain that the HermitianRandić energy of a mixed tree is the same as the Randić energy of its underlying graph.
Our future work will focus more on the characterizations of the HermitianRandić matrix of mixed graphs, such as the HermitianRandić spectrum of a complete mixed graph, more bounds on the HermitianRandić energy of mixed graphs with other parameters and mixed graphs that share the same HermitianRandić spectra with their underlying graphs.
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Acknowledgements
This work is supported by the National Natural Science Foundations of China (No.11171273).
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Lu, Y., Wang, L. & Zhou, Q. HermitianRandić matrix and HermitianRandić energy of mixed graphs. J Inequal Appl 2017, 54 (2017). https://doi.org/10.1186/s1366001713298
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DOI: https://doi.org/10.1186/s1366001713298
MSC
 05C50
 05C07
 05C31
Keywords
 mixed graph
 Hermitianadjacency matrix
 HermitianRandić matrix
 HermitianRandić energy