Omega Index of Line and Total Graphs

A derived graph is a graph obtained from a given graph according to some predetermined rules. Two of the most frequently used derived graphs are the line graph and the total graph. Calculating some properties of a derived graph helps to calculate the same properties of the original graph. For this reason, the relations between a graph and its derived graphs are always welcomed. A recently introduced graph index which also acts as a graph invariant called omega is used to obtain such relations for line and total graphs. As an illustrative exercise, omega values and the number of faces of the line and total graphs of some frequently used graph classes are calculated.


Introduction
Let G be a simple graph with V(G) � v 1 , v 2 , . . . , v n and E(G) � e 1 , e 2 , . . . , e m as the vertex and edge sets. n and m are called the order and size of G, respectively, and are the most important graph parameters. If e � uv ∈ E(G), we say that u and v are adjacent and e is incident to u and v. e number of edges incident to a vertex v is called the degree of v and denoted by d G v, or by dv if there is no confusion. A vertex of degree 1 is named as the pendant vertex. e set of degrees, of all vertices where Δ is the biggest vertex degree in G, is called the degree sequence of the graph. A graph which is connected and has no cycles is called a tree. A graph is called acyclic, unicyclic, bicyclic, tricyclic, etc. according to the number of cycles it has as 0, 1, 2, 3, etc. As usual, the path, cycle, star, complete, complete bipartite, and tadpole graphs are denoted by P n , C n , S n , K n , K r,s , and T r,s , respectively. For other graph theoretical notions used in this paper, see, e.g., [1][2][3].
Given a graph G, the line graph L(G) of G is the graph whose vertex set is E(G) with two vertices of L(G) being adjacent iff corresponding edges in G are adjacent. For some applications of the line graph, see, e.g., [4,5]. Similarly, the total graph T(G) of G is the graph whose vertex set is V(G) ∪ E(G) with two vertices of T(G) being adjacent iff the corresponding elements of G are either adjacent or incident. Line graphs and total graphs are two examples of derived graphs. Minimal doubly resolving sets and strong metric dimension of the layer sun graph and the line graph of this graph are calculated in [4]. e classical meanness property of some graphs based on line graphs was considered in [5]. For some recent applications of the total graphs, see, e.g., [6][7][8].
According to definitions, the degree sequences of the line and total graphs are (2)

Omega Index and Fundamentals
In this paper, we study the line and total graphs in relation with omega index and the number of faces known as the cyclomatic number. Omega index is an additive quantity defined for a given degree sequence (1) or for a graph with It is shown that Ω(G) � 2(m − n) and therefore it is always an even number. It is shown that the omega characteristic gives us very powerful information about cyclicness and connectedness of all the realizations of a given degree sequence (see, e.g., [9]). In brief, it is shown that all realizations of a degree sequence D with Ω(D) ≤ − 4 must be disconnected; each connected realization of a degree sequence D with Ω(D) � −2 must be a tree; each connected realization of a degree sequence D with Ω(D) � 0 must be a unicyclic graph; each connected realization of a degree sequence D with Ω(D) � 2 must be a bicyclic graph, etc. Also, the number of faces of a graph or all the realizations of a given degree sequence is formulated as where c(G) is the number of components of G. For more properties of the omega index, see [10,11]. e effect of edge and vertex deletion on the omega index is studied in [9]. Next, we obtain the number of pendant vertices of a caterpillar tree which consists of a main path so that all vertices are having maximum distance 1 from the path.
where d G i is the degree of v i .
Proof. By counting, v 1 has only one nonpendant neighbor and hence d G 1 − 1 pendant neighbors, v k has only one nonpendant neighbor and hence d G k − 1 pendant neighbors, and similarly, for i � 2, 3, . . . , k − 1, the vertex v i has d G i − 2 pendant neighbors. erefore, thus giving the result.

Corollary 1. For any tree T, we have
For a caterpillar tree, the degree sequence of its line graph can be stated more deterministically.

Theorem 2. Let T be a caterpillar tree. Let the degrees of the nonpendant vertices of
Proof. By eorem 5 in [11], there exists a complete graph Proof. By definition, we have □ In the following result, we calculate the omega index of the line graph of G when G is k-cyclic by means of triangular numbers T n � n(n + 1)/2.
e following results are about the omega index of the line graph.
□ Now, we obtain some results on the omega index of the total graphs.
Proof. It follows by the definition of omega index.
In [11], the number of faces of the line graph of a tree T was given by where T n � n(n + 1)/2 is the n − th triangular number. In [?], this number for a tricyclic graph G was given by ese suggest the following generalization.

Theorem 4. Let G be a simple, connected, and k − cyclic graph with degree sequence (1). e number of faces of the line graph L(G) is
Proof. For acyclic part of G, the formula is given in equation (20). For each t − cycle C in G, L(G) has another t-cycle C ′ formed by joining the midpoints of the edges of C. Hence, the number of cycles in G must be added to Δ i�3 a i T i−2 giving the result. □ Inverse problems in mathematics are quite important due to their applications. In graph theory, the inverse problem is the one which deals with finding the values of a given topological graph index. Here, we solve a similar problem for the number of faces of the line graph.

Theorem 5. Let G be a connected graph. en, r(L(G)) can take any positive integer value.
Proof. By eorem 4, we have equation (22) for a simple, connected, k-cyclic graph G.

Corollary 8. Let T be a tree with no vertices of degree 3. en, r(L(T)) can take all positive integers except
(24) As a i ≥ 0 are integers, the result follows.

Corollary 10. We have
(26) e following variation of this result has useful applications related to cyclicness.

Corollary 11. We have
By Corollary 11, we have the following cases:

Omega and r of the Line Graphs of Some Special Graphs
Now, we consider the omega and r values of some frequently used graph classes. First, we give a new proof of the fact that the line graph of P n is P n−1 .

Lemma 2.
We have L P n � P n−1 .

Omega and r of the Total Graphs of Some Special Graphs
In this section, we calculate the omega and r values of some frequently used graphs. We first give an important property. Proof. Let v ∈ V(G). Let e be an edge incident to v. As G is connected, there is at least one adjacent vertex say u, to v. In the total graph T(G), the vertex v will be adjacent to u and e implying the result.
□ e proof alternatively follows from the fact that DS(T(G)) consists of integers in the form of either 2d i or e j + 2, where d i , e j ≥ 1.
Next, we give the relation between omega of G and omega of T(G).

Theorem 6. For a connected graph G, we have
Proof. Recall that DS(T(G)) consists of a i times 2d i for every v i ∈ V(G) and de j + 2's for every e j � u j v j ∈ E(G). As de j � du j + dv j − 2, we can deduce that DS(T(G)) consists of a i times 2d i for every v i ∈ V(G) and du j + dv j 's for every e j � u j v j ∈ E(G). Hence, � Ω(G) + M 1 (G).
Finally, we give the following result for the omega indices of the total graphs of some frequently used graph classes: e omega index of the total graphs of some well-known graph classes is as follows: Ω T P n � 4n − 8, Ω T C n � 4n, Ω T S n � (n + 1)(n − 2), Ω T K n � n(n + 1)(n − 2), Ω T K r,s � (r + s)(rs − 2) + 2rs, Ω T T r,s � 4(r + s) + 2. (37)

Conclusion
Derived graphs are graphs obtained from a given graph according to some rules. In this paper, two of the most frequently used derived graphs, the line and total graphs, are studied. Calculating some properties of a derived graph helps to calculate the same properties of the original graph.
Here, by means of omega index and several results in recent papers, new relations for line and total graphs are obtained. Also, omega values and the number of faces of the line and total graphs of some frequently used graph classes are calculated. In the future works, similar ideas can be applied to establish several relations for other derived graphs.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.