On First Hermitian-Zagreb Matrix and Hermitian-Zagreb Energy

A mixed graph is a graph with edges and arcs, which can be considered as a combination of an undirected graph and a directed graph. In this paper we propose a Hermitian matrix for mixed graphs which is a modified version of the classical adjacency matrix related to the first Zagreb index. We also establish some results related to this newly proposed matrix. Finally we present some results on the Hermitian-Zagreb energy. Keywords— first Zagreb index, Hermitian adjacency matrix, energy. MSC: 05C50; 05C07; 05C31


I INTRODUCTION
In chemical graph theory, topological index or molecular descriptor is a numerical value associated with chemical constitution for correlation of chemical compounds with various physio-chemical properties (for example: boiling point, enthalpy of vaporization, stability and so on) or biological activities. Over the period of last few decades, many topological indices were defined and studied based on degree, distance between the vertices etc. An account of some degree-based topological indices may be found in [8]. In a paper published in 1972, Trinajsti and Gutman [7] discussed the dependency of total electron π-energy of alternant hydrocarbons on molecular structure in the form of a graph G and found two terms that occur in its approximate expressions which are of the form: is the degree of the vertex u in G. in case of mixed graph M it is the degree of u in the underlying graph of M. Immediately, they were proposed to measure the branching of carbon-atom skeleton [6] for their proportional behaviour with the extent of branching. Few years later only, Balaban et al., [1] used the terminology first and second Zagreb indices' (more precisely Zagreb group indices) for M 1 and M 2 respectively and called them as topological indices.
In 2005, Rodr ǵuez initiated the study of spectral properties of topological indices by associating weighted adjacency matrices to the indices [13]. Since then a reasonable number of works were reported on the study of spectral properties of the topological indices [2], [3], [14], [15]. Recently in 2018, Rad et al., communicated some results on Zagreb energy and Zagreb Estrada index of graphs [12]. In these studies researchers consider only undirected graphs, but in last few years some attempts are also reported for oriented as well as mixed graphs for Randi ́ index [4], [11].
The Hermitian-adjacency matrix of a mixed graph  [5]. From the above we can see that if we add a first Zagreb 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? This motivates us to define the first Hermitian-Zagreb matrix, ( ) of a mixed graph M. Let us denote and arc by (p, q) and an edge (undirected or both directed) by simply pq. The matrix ( ) can be defined as follows: In 1978, Gutman introduced the concept of graph energy [9]. Since then many extensions of this graph invariant have been proposed. In this paper we also define an extension of this graph invariant based on the eigenvalues of the above proposed first Hermitian-Zagreb matrix. The rest of the paper is organized as follows. In the next section we study the spectral properties of the first Hermitian-Zagreb matrix. In section 3, we report some results on the first Hermitian-Zagreb energy. In section 4, conclusions are made.

II PROPERTIES OF THE FIRST HERMITIAN-ZAGREB MATRIX
In this section first we analyze the matrix ( ) in terms of its structure and spectral properties. It is clear that In the above expression we have considered only those values of l and k for which either (l, k) or (k, l) is an arc, or lk is an edge. This is because, otherwise ( ) , and it is not going to contribute anything to the summation. If (l, k)( or (k, l)) is an arc then ( ) ( ) (or ( )). Again if lk is an edge, then ( ) ( ). In all the cases it is easy to verify that ( ) ( ) ̅̅̅̅̅̅̅̅ ( ) . And hence ∑ ∑ ( ( ) ) = 2 HZ ( ). The last part (c) can be easily proved by using the identity (∑ ) ∑ + ∑ Theorem 2.2: Let λ 1 , λ 2 , λ 3 , …, λ n be the eigenvalues of ( ) and = . Then }, each of these sets contains (n-1) elements. We have , On simplification we have the result. Now we will give an expression of determinant and the characteristic polynomial of ( ) To obtain the expression we first need to introduce some basic definitions. The value of a mixed walk W=v 1 v 2 …v k is W= (W)= ( ) ( ) ( ) ( ) ( ) A mixed walk is called positive (negative) if this value for that walk is positive (negative). For sign we will consider the sign of the value ignoring real or complex. 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 ̅. 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 edge-component 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 [11]. The proofs of these results are avoided because they are similar to the ones done by Lu et al., in [11].

IV CONCLUSIONS
In this paper it is tried to extend the classical adjacency matrix for mixed graphs, corresponding to first Zagreb index and then we study this matrix for its spectral properties. An extension of the graph invariant "graph energy" is also proposed. We call it "first Hermitian-Zagreb energy". We establish few bounds of this novel extension of graph energy. In future work we will try to study similar matrix for the second Zagreb index. We also plan to study this matrix for oriented graphs for which we commonly define a skew symmetric matrix.