Hermitian-Randić matrix and Hermitian-Randić energy of mixed graphs

Let M be a mixed graph and H(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H(M)$\end{document} be its Hermitian-adjacency matrix. If we add a Randić weight to every edge and arc in M, then we can get a new weighted Hermitian-adjacency matrix. What are the properties of this new matrix? Motivated by this, we define the Hermitian-Randić matrix RH(M)=(rh)kl\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{H}(M)=(r_{h})_{kl}$\end{document} of a mixed graph M, where (rh)kl=−(rh)lk=idkdl\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(r_{h})_{kl}=-(r_{h})_{lk}=\frac{\mathbf{i}}{\sqrt {d_{k}d_{l}}}$\end{document} (i=−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbf{i}=\sqrt{-1}$\end{document}) if (vk,vl)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(v_{k},v_{l})$\end{document} is an arc of M, (rh)kl=(rh)lk=1dkdl\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(r_{h})_{kl}=(r_{h})_{lk}=\frac{1}{\sqrt{d_{k}d_{l}}}$\end{document} if vkvl\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$v_{k}v_{l}$\end{document} is an undirected edge of M, and (rh)kl=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(r_{h})_{kl}=0$\end{document} otherwise. In this paper, firstly, we compute the characteristic polynomial of the Hermitian-Randić matrix of a mixed graph. Furthermore, we give bounds on the Hermitian-Randić energy of a general mixed graph. Finally, we give some results about the Hermitian-Randić 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  , v  , . . . , v n } and edge set E(M). For v i , v j ∈ 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 ↔ v j ). Denote a directed edge (or arc) from v i to v j by (v i , v j ) (or v i → v j ). In addition, let E  (M) denote the set of all undirected edges and E  (M) denote all the directed arcs set. Clearly, E(M) is the union of E  (M) and E  (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 [] for terminologies and notations not defined here.
Let G be a simple graph with vertex set {v  , v  , . . . , v n }. The adjacency matrix of a simple graph G of order n is defined as the n × n symmetric square matrix A = A(G) = (a ij ), where a ij =  if v i v j is an edge of G, otherwise a ij = . We denote by A convenient parameter of G is the general Randić index R α (G) defined as R α (G) = uv∈E(G) (d u d v ) α , where the summation is over all (unordered) edges uv in G. The molecular structure-descriptor, first proposed by Randić [] in , is defined as the sum of  Gutman et al. [] pointed out that the Randić-index-concept is purposeful to associate the graph G with a symmetric square matrix of order n, named Randić matrix An oriented graph G σ is a digraph which assigns each edge of G a direction σ . The skew is defined as the sum of the norms of all the eigenvalues of S(G σ ). For more details about skew energy, we can refer to [, ].
In , Gu, Huang and Li [] defined the skew Randić matrix R s (G σ ) = ((r s ) ij ) of an The Hermitian-adjacency matrix of a mixed graph M of order n is the n × n ma- otherwise.
Let In this paper, we define the Hermitian-Randić matrix of a mixed graph M and study some basic characteristics of the Hermitian-Randić matrix of mixed graphs. In Section , we give the characteristic polynomial of the Hermitian-Randić matrix of a mixed graph M.
In Section , we study some bounds on the Hermitian-Randić energy of mixed graphs with different parameters and give the conditions under which mixed graphs can attain those Hermitian-Randić energy bounds. In Section , we show that the Hermitian-Randić energy of a mixed tree is the same as the Randić energy of its underlying graph. In Section , we summarize the results of this paper and give some future works we will study.

Hermitian-Randić characteristic polynomial of a mixed graph
In this section, we will give the characteristic polynomial of a Hermitian-Randić 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 ). 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 α.
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).
where S n is the set of all permutations on {, , . . . , n}.
is not an edge or arc of M, then (r h ) kπ (k) = ; that is, this term vanishes. Thus, if the term corresponding to a permutation π is non-zero, then π is fixed-point-free and can be expressed uniquely as the composition of disjoint cycles of length at least .
) and vice versa. Thus, they cancel each other in the summation. In addition, if for some direction of a permu- , then for the other direction C  has the same value. For each edge-component (kl) corresponding to the factors ( For each arc-component (kl) corresponding to the factors = x n + a  x n- + a  x n- + · · · + a n , then the coefficients of P R H (M, x) are given by Proof The proof follows from Theorem . and the fact that (-) k a k is the summation of determinants of all principal k × k submatrices of R H (M).

Bounds on the Hermitian-Randić energy of mixed graphs
In this section, we will give some bounds on the Hermitian-Randić energy of mixed graphs. First, we will give some properties of a Hermitian-Randić matrix of mixed graphs.
Lemma . Let M be a mixed graph of order n ≥ .
From Lemma ., we can obtain the following theorem. So, This completes the proof.
Similar to Theorem ., we can obtain the following theorem. On the other hand, By using an arithmetic geometric average inequality, we can get that Therefore, we can obtain the lower bound on the Hermitian-Randić energy From the Cauchy-Schwarz 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 |μ  | = |μ  | = · · · = |μ n |, i.e., there exists a constant c = |μ i |  for all i such that R  H (M) = cI n . This completes the proof.

with equalities holding both in the lower bound and upper bound if and only if
Proof If M is a mixed graph and its underlying graph M U is r regular, then R - (M U ) = m r  and m = nr. By Theorems . and ., we can obtain the results.

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 () n is even and G is the disjoint union of n/ paths of length , or () n is odd and G is the disjoint union of (n -)/ paths of length  and one path of length .
Combining Theorem . and Lemma ., we can get upper and lower bounds for the Hermitian-Randić energy by replacing R - (M U ) with other parameters. We now give bounds of the Hermitian-Randić energy of a mixed graph with respect to its order.
Theorem . Let M be a mixed graph of order n ≥  without isolated vertices and M U be its underlying graph. Let μ  ≥ μ  ≥ · · · ≥ μ n be the Hermitian-Randić 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/ paths of length . The equality in the lower bound holds if and only if M U is a complete graph and μ
From the definition of the Hermitian-Randić energy of a mixed graph, we have Combining this with Lemma ., we have From the proof above and Lemma ., we know that the equality in the lower bound holds if and only if M U is a complete graph and μ k μ l ≥  or μ k μ l ≤  for all  ≤ k < l ≤ n. Note that n k= μ k =  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 μ  = -μ n = , μ j = , j = , . . . , n -.
This completes the proof.
Remark . It should be pointed out that when M is a complete mixed graph, its Hermitian-Randić spectrum is not unique. For example, let M U = K  , if all edges of E(M) are oriented, then we have μ  = -μ  = √   , μ  = , then we can obtain the lower bound in Theorem .. If some edges of E(M) are undirected, then we can not obtain the lower bound in Theorem .. For example, if (r h )  = (r h )  = i  , (r h )  =   , then μ  =  and μ  = μ  = -  . Hence, the problem of determining all complete mixed graphs for which the lower bound in Theorem . is attained appears to be somewhat more difficult.
To deduce more bounds on E R H (M), the following lemma is needed.

Now we turn to new bounds on E R H (M).
Theorem . Let M be a mixed graph of order n and M U be its underlying graph. Let μ  ≥ μ  ≥ · · · ≥ μ n be the Hermitian-Randić spectrum of R H (M). Then Proof Note that It follows that This together with () implies that So, This completes the proof.
Note that the right-hand side of () is a non-decreasing function on α ≥ . Combining this with Theorem ., we have the following corollary.
Corollary . Let M be a mixed graph of order n and M U be its underlying graph. Let μ  ≥ μ  ≥ · · · ≥ μ n be the Hermitian-Randić spectrum of R H (M). Then In particular, if M is a connected mixed bipartite graph, then we have the following theorem.
Theorem . Let M be a connected mixed bipartite graph of order n and M U be its underlying graph. Let μ  ≥ μ  ≥ · · · ≥ μ n be the Hermitian-Randić spectrum of R H (M). Then Proof Note that M U is a bipartite graph. By Corollary .(), we have μ i = -μ n+-i and μ i ≥  for i = , , . . . , n  . Therefore, It follows that This together with () implies that So, This completes the proof.
Note that the right-hand side of () is a non-decreasing function on α ≥ . Combining this with Theorem ., we have the following corollary.
Corollary . Let M be a connected mixed bipartite graph of order n and M U be its underlying graph. Let μ  ≥ μ  ≥ · · · ≥ μ n be the Hermitian-Randić spectrum of R H (M). Then

Hermitian-Randić energy of trees
In [], the authors proved that the skew energy of a directed tree is independent of its orientation. In [], the authors showed that the skew Randić energy of a directed tree has the same result. In this section, we will show that the Hermitian-Randić energy also has the same result. In the beginning of this section, we first characterize the mixed graphs with cut-edge. Thus, the Hermitian-Randić spectrum and the Hermitian-Randić energy are invariants when reversing the cut-arc's orientation or unorienting it or orienting an undirected cutedge. By applying Theorem ., we can obtain the following corollaries.

Corollary . Let T be a mixed tree and T U be its underlying graph. Then
() The Hermitian-Randić energy of T is independent of its orientation of the arc set.
() The Hermitian-Randić energy of T is the same as the Randić energy of T U .

Conclusions
In this paper, we define the Hermitian-Randić matrix of a mixed graph M and give the definitions of Hermitian-Randić characteristic polynomial and Hermitian-Randić energy of a mixed graph M. We give the bounds on the Hermitian-Randić energy of a mixed graph M with respect to its order, the Hermitian-Randić spectrum and a general Randić index (with α = -). We also obtain that the Hermitian-Randić 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 Hermitian-Randić matrix of mixed graphs, such as the Hermitian-Randić spectrum of a complete mixed graph, more bounds on the Hermitian-Randić energy of mixed graphs with other parameters and mixed graphs that share the same Hermitian-Randić spectra with their underlying graphs.