On ve -Degree Irregularity Indices

: In this paper, vertex - edge degrees (or simply, ve - degrees) of vertices in a graph are considered. The ve - degree of a vertex v in a graph equals to the number of different edges which are incident to a vertex from the closed neighborhood of v . The author introduces the ve -degree total irregularity index here and calculates this index for paths and double star graphs. Finally, the maximal trees are characterized with respect to the ve - degree total irregularity index.


INTRODUCTION
HE examination of molecular structures expressed through graphs is one of the important pillars of graph applications. In an undirected graph, the degree sequence is a uniform sequence of the degrees of its vertices that does not increase. Invariants belonging to graphs are most commonly referred to as topological indices and they are often stated using the degrees of vertices, distances between vertices, eigenvalues, symmetries, and many other properties of the graphs. The term topological index first appeared in a study by Wiener. [1] Topological indices lead us to foretell particular physico-chemical properties such as boiling point, melting point, enthalpy of evaporation, stability etc. There are more than 150 topological indices currently known and used. Among these indices, degree-based topological indices are more remarkable and they are quite handy tools for chemists. For more information on degree-based topological indices, I refer to the paper, [2] which is a detailed review article on this. Graphs are one of the basic tools used in the studies conducted in many mathematical sciences. [3][4][5] An organic compound and its molecular structure are usually indicated by a molecular graph. Here, atoms imply vertices, and bonds between atoms imply edges. Thus, an idea about the physical properties of these chemical compounds is obtained. Today, chemical graph theory studies are a discipline that has an important place in the fields of chemistry, biology, electrical networks and drug designs. Investigation of compounds with the same chemical formula, even if their chemical structures are different, is the field of study of this discipline. There are many important and remarkable conclusions regarding chemical indices for the studies of computational complexity and chemical graph theory. [6] Albertson index is one of the most important topological indices and it was introduced in 1997. [7] Consider a simple and finite graph. Let G be this graph with the set of vertices V(G) and the set of edges E(G). The degree of a vertex u of the graph G is the number of adjacent vertices with u in G and it is indicated by deg( ). u A graph G is called regular if all its vertices have the same degree. A graph that is not regular is called irregular. Albertson stated the graph invariant as and named it as irregularity of the graph G. In other words, the Albertson index and irregularity mean the same thing. He obtained some upper bounds for trees, bipartite graphs and triangle-free graphs in his study. [7] Graphs with the maximal irregularity were characterized by Abdo et al. They took a different approach than Albertson and found a sharp upper bound for graphs with n vertices and some lower bounds on the maximal irregularity of graphs. [8] Also, the total version of the Albertson index was recently defined by Abdo et al. They determined all graphs with maximal total irregularity. [9] A comparison between the irregularity and total irregularity was made in [10] and some inequalities were obtained for connected graphs and trees. Moreover, some well-known irregularity measures were compared [11] and it was shown that any two of these irregularity indices are mutually inconsistent. This means that it is difficult to decide definitively which is more and which is less irregular. Gutman demonstrated the calculations of the irregularity measures on molecular graphs and made comparisons between these results. [12] The trees which were the most and least irregular were characterized according to the Albertson index. [13] The irregularity measure based on eigenvalues of graphs described by Collatz and Sinogowitz [14] has been the oldest known numerical irregularity measure. Bell introduced a second such measure based on the variance of the vertex degrees of a graph, another irregularity measure. [15] He determined the most irregular graphs with respect to these two measures. More details about the irregularity of the graphs can be found in the book. [16] Domination is one of the most important graph invariants. A subset ( ) D V G ⊆ is a dominating set, if every vertex in G either is an element of D or is adjacent to at least one member of D. The domination number is the number of vertices in a smallest dominating set for G. [17] Domination has been shown to be a very sensitive graph theoretical invariant to even the slightest changes in a graph. [18] Domination was studied for chemical materials in the past. For example, the domination number of benzenoid chains and hexagonal grid was obtained by Vukičević and Klobučar. [18] Vertex-edge domination (ve-domination) and edgevertex domination (ev-domination) are two mixed type domination invariants. An edge e dominates a vertex v, if e is incident to v or e is incident to a vertex which is adjacent is an edge-vertex dominating set of a graph G, if every vertex of G is ev-dominated by at least one edge of D. The minimum cardinality of an evdominating set is called the ev-domination number. A vertex v ve-dominates an edge e which is incident to v and any edge which is adjacent to e. A set The minimum cardinality of a ve-dominating set is called the ve-domination number. The ve-domination and ev-domination concepts were introduced by Peters. [19] The lower and upper bounds on the ve-domination and ev-domination numbers in different graphs were studied. [20] Also, total edge-vertex domination was introduced recently. [17] Chellali et al. introduced two degree concepts: ve-degree and ev-degree of the graphs based on ve-domination and ev-domination. [21] The ve-degree of a vertex ( ) v V G ∈ equals the number of edges ve-dominated by v. The ev-degree of an edge e uv = equals the number of vertices ev-dominated by e. The regularity and irregularity of graphs about ve-degree and ev-degree were studied by Horoldagva et al. [22] A graph is ve-regular if all its vertices have the same ve-degree. A graph is ev-regular if all its edges have the same ev-degree.
The ve-degree and ev-degree concepts of graphs were widely applied to Chemical Graph Theory. [23,24] Many papers were written about the modified versions of the various topological indices with respect to ve-degree and ev-degree. Some chemical materials were investigated with these modified versions of the topological indices. For example, the ve-degree and ev-degree based topological properties of single walled titanium dioxide nanotube, [25] h-naphtalenic nanotube, [26] silicon carbide Si2C3-II[p,g], [27] two carbon nanotubes, [28] polycyclic graphite carbon nitride [29] and crystallographic structure of cuprite Cu2O [30] were studied. It has been seen that ve-degree and evdegree topological indices can be used as possible tools in QSPR researches.
The ve-degree irregularity index was defined recently. [31] The definition of this concept is presented in the second section. Moreover, the maximal trees were characterized with respect to this index. [31] In this paper, I define the ve-degree total irregularity index and compute this index for paths and double star graphs. Finally, I obtain the maximal trees with respect to the ve-degree total irregularity index.

PRELIMINARIES
Let G be a simple graph with the vertex set V(G) and the edge set E(G) such that ( ) V G n = and ( ) .
the open neighborhood of u is defined as The degree of a vertex u is the cardinality of ( ) G N u and it is denoted by deg( ).
u A vertex which has degree one is called a leaf. The ve-degree of a vertex v equals to the number of different edges which are incident to a vertex from the closed neighborhood of v and it is denoted by deg ( ).
ve v Moreover, the ev-degree of an edge e = ab equals to the number of vertices of the union of the closed neighborhoods of a and b, it is denoted by deg ( ).
ev v A graph G is ve-regular if all its vertices have the same ve-degree. The paths, cycles, complete graphs and stars of order n are denoted by It is known that 1, 1 n S − is ve-regular tree such that all its vertices have same ve-degree 1. n − [21] The cycle ( 4) n C n ≥ is the unique unicyclic graph which is ve-regular. [21] For simplicity, a ve-regular graph, each of whose vertices has ve-degree r, is called rve-regular. [22] For example, the cycle graph is 4ve-regular for 4 n ≥ . Furthermore, the complete graph Kn is mve-regular such that the size -1 / 2. ( ) m n n = Definition 2.1. For a connected graph G, Definition 2.2. For a connected graph G, such that G η is the total number of triangles contained in G. [21] It implies that for a triangle-free graph G, Definition 2.6. Let G be a graph of order n. Then, the total irregularity index of G is computed by If the degrees of vertices are ordered as the total irregularity index can be calculated by If T is a tree of order n, then such that the equality holds if and only if T is a star. [13] ) such that the equality holds if and only if T is a star. [9] The ve-degree version of the Albertson index was expressed as in the following definition. Definition 2.9. Let G be a graph of order n. Then, the vedegree total irregularity index of G is computed by If the ve-degrees of vertices are ordered as the ve-degree total irregularity index can be calculated by The repeated degrees can be shown by exponential numbers.   Proof. In order to prove the equalities, I apply some operations to 1, 1 n S − graphs. It is clear that the star graphs are ve-regular graphs. Then, the ve-degree irregularity index and ve-degree total irregularity index of stars equal to 0. 18 0 for n 7. n + > ≥ If the operation which is used in the transformation from 1 T to 2 T is used (n -3)-times to a star, I obtain a path at the end. Now I remove a leaf s and attach it to a vertex (r) which is incident to the central vertex u on 1 .

MAIN RESULTS
T Thus, I obtain the tree 3 T , which is indicated in Figure 1.   It implies that 6 T has the maximal ve-degree total irregularity index of even order in the trees.

CONCLUSION
After the introduction of the ve-degree irregularity index, [31] the ve-degree total irregularity index is defined in this paper. Moreover, the ve-degree total irrregularity index of paths and double star graphs are obtained, and the maximal graphs with respect to this index are attained. Consequently, the present paper is a contribution to find the ve-degree based topological indices in different sciences. By means of the ve-degree based topological indices, the number of tools which are used in the computation of graph irregularity is increased.
As the paralleling of the rapid growing of science and technology, the importance of analysing in networks is increased. Then, the ve-degree irregularity indices may be used in the computation of the chemical, biological and other properties of chemical materials.