Abstract
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.
1. Introduction
Let be a simple graph with and as the vertex and edge sets. and are called the order and size of , respectively, and are the most important graph parameters. If , we say that and are adjacent and is incident to and . The number of edges incident to a vertex is called the degree of and denoted by , or by if there is no confusion. A vertex of degree 1 is named as the pendant vertex. The set of degrees,of all vertices where is the biggest vertex degree in , 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 , and , respectively. For other graph theoretical notions used in this paper, see, e.g., [1–3].
Given a graph , the line graph of is the graph whose vertex set is with two vertices of being adjacent iff corresponding edges in are adjacent. For some applications of the line graph, see, e.g., [4, 5]. Similarly, the total graph of is the graph whose vertex set is with two vertices of being adjacent iff the corresponding elements of 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]. The 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., [6–8].
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 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 with must be disconnected; each connected realization of a degree sequence with must be a tree; each connected realization of a degree sequence with must be a unicyclic graph; each connected realization of a degree sequence with 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 aswhere is the number of components of . For more properties of the omega index, see [10, 11]. The 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.
Theorem 1. Let be a caterpillar tree. Let the nonpendant vertices of be so that has unique nonpendant neighbor ; has unique nonpendant neighbor ; and has two nonpendant vertices and for . The following relation holds:where is the degree of .
Proof. By counting, has only one nonpendant neighbor and hence pendant neighbors, has only one nonpendant neighbor and hence pendant neighbors, and similarly, for , the vertex has pendant neighbors. Therefore,thus giving the result.
Corollary 1. For any tree , we have
For a caterpillar tree, the degree sequence of its line graph can be stated more deterministically.
Theorem 2. Let be a caterpillar tree. Let the degrees of the nonpendant vertices of be . The degree sequence of the line graph of is
Proof. By Theorem 5 in [11], there exists a complete graph in around each nonpendant vertex of degree in . Also, iff and have a common vertex. Therefore, has vertices of degree ; has vertices of degree ; and each of other complete graphs for has vertices of degree . Also, the intersection vertex of and is of degree for . Hence, the result follows.
The next result gives the relation between the edge numbers of and by means of the first Zagreb index of :
Lemma 1. For a given graph , we have
Proof. By definition, we have
In the following result, we calculate the omega index of the line graph of when is -cyclic by means of triangular numbers .
Corollary 2. For a given graph with , we have
The following results are about the omega index of the line graph.
Theorem 3. For a graph , we have
Proof. As and , we have
Corollary 3. For a graph , we have
Proof. By Theorem 3,thus giving the result.
Corollary 4. For a graph , we have
Proof. It follows from .
Now, we obtain some results on the omega index of the total graphs.
Corollary 5. For a connected graph , we have
Corollary 6. For a connected graph , we have
Proof. It follows by Corollary 5 and Theorem 3.
Corollary 7. For any graph , we have
Proof. It follows by the definition of omega index.
This means that for every graph , is fixed and is equal to .
In [11], the number of faces of the line graph of a tree was given bywhere is the triangular number. In [?], this number for a tricyclic graph G was given by
These suggest the following generalization.
Theorem 4. Let be a simple, connected, and graph with degree sequence (1). The number of faces of the line graph is
Proof. For acyclic part of , the formula is given in equation (20). For each in , has another -cycle formed by joining the midpoints of the edges of . Hence, the number of cycles in must be added to 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 be a connected graph. Then, can take any positive integer value.
Proof. By Theorem 4, we have equation (22) for a simple, connected, -cyclic graph . This gives us the following linear equality:As are integers and as has coefficient 1, can take any positive integer value.
Some special cases are as follows.
Corollary 8. Let be a tree with no vertices of degree 3. Then, can take all positive integers except 1, 2, 4, 5, 7, 8, 11, 14, and 17.
Proof. By the equality in Theorem 11 in [12], we haveAs are integers, the result follows.
Corollary 9. Let be a tree with no vertices of degree 3 or 4. Then, can take all positive integer values except , and 29.
Proof. Note thatAfter easy observations, we find that the class has all integers ending with 0; has all integers ending with 6; has all integers ending with 2; has all integers ending with 8; has all integers ending with 1; has all integers ending with 3; has all integers ending with 4; has all integers ending with 5; has all integers ending with 7; and finally, has all integers ending with 9. This implies that only the values given above cannot be attainable by .
Then, we obtain the following result.
Corollary 10. We have
The following variation of this result has useful applications related to cyclicness.
Corollary 11. We have
By Corollary 11, we have the following cases:(i)If is acyclic, then (ii)If is unicyclic, then (iii)If is bicyclic, then (iv)If is -cyclic, then
3. Omega and of the Line Graphs of Some Special Graphs
Now, we consider the omega and values of some frequently used graph classes. First, we give a new proof of the fact that the line graph of is .
Lemma 2. We have
Proof. Recall that . We then have
Hence, as path graphs are acyclic. Also, .
Secondly, it is obvious that . Therefore, and .
Next, we consider the line graph of the star graph . By the definition of the line graph, . Hence,
For a complete graph , we have implying that and . In Figure 1, we illustrated the case of .
For a complete bipartite graph , is a regular graph of degree . Its degree sequence is . Hence, and .
Finally, consider the tadpole graph . has degree sequence . Hence, , and therefore, .
4. Omega and of the Total Graphs of Some Special Graphs
In this section, we calculate the omega and values of some frequently used graphs. We first give an important property.
Lemma 3. Let be a connected graph. Then, cannot have any pendant vertex.
Proof. Let . Let be an edge incident to . As is connected, there is at least one adjacent vertex say , to . In the total graph , the vertex will be adjacent to and implying the result.
The proof alternatively follows from the fact that consists of integers in the form of either or , where .
Next, we give the relation between omega of and omega of .
Theorem 6. For a connected graph , we have
Proof. Recall that consists of times for every and ’s for every . As , we can deduce that consists of times for every and ’s for every . Hence,
Corollary 11. For a connected graph , we have
Proof. As in Theorem 6, we can write
Corollary 12. If is an acyclic graph, thenand if is a unicyclic graph, then
Finally, we give the following result for the omega indices of the total graphs of some frequently used graph classes:
Theorem 7. The omega index of the total graphs of some well-known graph classes is as follows:
5. 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
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to this paper.