Abstract
Average degree of a graph is defined to be a graph invariant equal to the arithmetic mean of all vertex degrees and has many applications, especially in determining the irregularity degrees of networks and social sciences. In this study, some properties of average degree have been studied. Effect of vertex deletion on this degree has been determined and a new proof of the handshaking lemma has been given. Using a recently defined graph index called index, average degree of trees, unicyclic, bicyclic, and tricyclic graphs have been given, and these have been generalized to -cyclic graphs. Also, the effect of edge deletion has been calculated. The average degree of some derived graphs and some graph operations have been determined.
1. Introduction
Let be a finite, undirected, and simple graph, having vertices and edges. For a vertex , the number of edges of meeting at is denoted by or and known as the degree of . If the set of all vertex degrees of iswhere s are nonnegative integers, then is called the degree sequence of . Here, is the maximum vertex degree. For every graph , there is a degree sequence. However, for a degree sequence, there may or may not exist a graph. If there is a graph, then is said to be realizable. For every realizable degree sequence, there is at least one graph; usually, there are many.
In [1], a new graph index called index was defined by
Several properties of index were obtained in [1, 2].
There are some graph invariants related to the vertex degrees. The density of a graph measures how many edges are there in the set of edges compared to the maximum possible number of edges between the vertices of . For a simple graph, the maximum number of edges is attained when the graph is complete. In this case, this number would be . So, the density of is
Clearly, the density increases proportionally with the number of edges. Therefore, amongst all connected simple graphs having vertices, trees have the lowest density and complete graph has the highest density which is 1.
In this paper, we make use of index to study the properties of average degree of a graph which measures how many edges has compared to the number of vertices, that is,
From the definitions, we can easily deduce the following relation between and :
Also, the following relations between the index of a graph and density and average degree can be deduced as follows:
If all the vertices in a graph have the same degree, say , the graph is said -regular. Most of the graphs are not regular and some graph indices have been defined to measure the irregularity of a graph. The Bell index of a graph is defined bysee [3], and the degree deviation index is defined by
In [4], the degree deviation is equal to the product of the order of with the discrepancy of the graph. Also, the Collatz–Sinogowitz index is defined bywhere denotes the largest eigenvalue of the adjacency matrix [5]. Note that, in all these irregularity indices, the average degree of a graph is used. This makes the notion of the average degree of a graph important. In the recent paper [6], two new and structurally different irregularity indices IRA and IRB have been defined and compared with other existing irregularity measures. Naturally, this list can easily be extended to have new members of the family of irregularity indices.
The structure of this paper is planned as follows. In Section 2, two related notions, density and average degree, are studied and these two quantities are calculated for acyclic, unicyclic, bicyclic, tricyclic, and, in general, -cyclic graphs. In Sections 3 and 4, the effects of vertex and edge deletion on average degree of a graph are formulized. In Sections 5 and 6, the average degree of some derived graphs and some binary graph operations are determined.
2. Density, Average Degree, and Cyclicness
A connected graph having no faces is called acyclic. A graph having at least one face is named as cyclic; especially as unicyclic, bicyclic, tricyclic, etc., according to the number of faces which is , etc., respectively. Here, using the index, graphs are classified according to their cyclicness, and for each case, the density of the graph is characterized as follows:(i)If is acyclic, then (ii)If is unicyclic, then (iii)If is bicyclic, then (iv)If is tricyclic, then (v)If is -cyclic, for , then
Indeed, let be a connected -cyclic graph. By Theorem 1 in [1], we have , as is connected. Hence, , and by Corollary 3.4 in [1], we know that .
The following is a similar result for an average degree. The average vertex degree of a connected -cyclic graph is
As a result, connected acyclic, unicyclic, bicyclic, tricyclic, etc., graphs have average degrees , 2, , , etc., respectively.
3. Effect of Vertex Deletion on Average Degree
Now, the effect of vertex deletion on average degree will be considered. Using the obtained formula successively, we give a result which helps to calculate the average degree of a large graph by means of the average degree of a much smaller graph. Let be the graph obtained by deleting the vertex together with edges incident to . First, we have the following.
Theorem 1. The following recurrence relation holds:
Proof. Let be a connected simple graph of order . Let have degree in . is the graph obtained by deleting the vertex together with edges incident to . Hence, deleting from reduces the degree of each of the neighbours of by 1. So, the deletion of from reduces the total vertex degree of by . Hence,which gives the result.
Applying Theorem 1 recursively, we can give another proof of the handshaking lemma by labeling the vertices so that .
Corollary 1. (handshaking lemma). In every graph, the sum of vertex degrees is equal to twice the number of edges, that is,
Proof. Applying Theorem 1 successively to delete from , respectively, we get the following equalities:Adding all these side by side, we obtain the result.
So, we deleted all the vertices one by one to prove handshaking lemma. What happens if we only delete some of the vertices? We answer this question in two steps.
Theorem 2. Let be a connected simple graph of order , size , and average degree . If are pairwise nonadjacent vertices of , thenwhere , for .
Proof. Let and be the size and order of . Then, by definition, we have . Clearly, as vertices are deleted. When a vertex is deleted, note that incident edges are also deleted. So, if we delete pairwise nonadjacent vertices from , a total of edges are also deleted giving that . It proves the result.
Finally, if some pairs of the deleted vertices are adjacent, then we would have the following result.
Theorem 3. If pairs of the vertices are adjacent in , then
Proof. The proof is quite similar to the one of Theorem 2, except we need to deal with adjacent pairs of vertices. If and are two adjacent vertices in , then the total number of neighbours of is and the total number of neighbours of is . Hence, when deleting both and , we also delete edges incident to at least one of or . Here, comes as the edge between and which can only be deleted once, but it is incident to both and . So, when are deleted from , a total of edges are also deleted, giving the result.
Example 1. Let be the graph in Figure 1, and let us delete , and .
Note that the average degree of is . The graph is in Figure 2 and has the average degree .
According to Theorem 3,as there are pairs of deleted vertices.
4. Effect of Edge Deletion on Average Degree
We now determine the effect of edge deletion from a graph on the average vertex degree. Let be a graph of order and size , and let be an edge. The graph obtained by deleting the edge from is denoted by . We have the following recurrence relation between the average degrees of and .
Theorem 4. The average degree of is related to the average degree of by the following recurrence relation:
Proof. As deleting an edge will reduce the number of edges by 1 and will not change the number of vertices, we know thatgiving the required result.
5. Average Degree of Some Derived Graphs
A derived graph is a graph obtained from a given graph after some operation. In that sense, many authors consider derived graphs as graph operations. Derived graphs help to determine a property of a given graph by calculating the same property of the derived graph. In this section, we determine the average degree of some derived graphs. The derived graphs under study are the line, total, jump, and semitotal line graphs.
The line graph of a graph is constructed as follows. For each edge in the graph , we draw a new vertex of so that two edges in , having a vertex in common, make an edge between their corresponding vertices in . It can be seen that the order and size of the line graph are and , respectively, where is the famous topological graph index called the first Zagreb index. Indeed,
Hence, the average degree of the line graph is obtained as
The total graph of a graph , also known as the generalization of the line graph, is the graph so that the vertex set of corresponds to the vertices and edges of and two vertices are adjacent in iff their corresponding elements are either adjacent or incident in . By the definition, one can deduce the order and size of the total graph as and , respectively. Hence, we deduce the average degree of the total graph as
Next, we study the average degree of the jump graph. The jump graph of a graph is the complement of the line graph. That is, the jump graph of a graph is the graph defined on the edge set of the graph in which two vertices are adjacent if and only if they are not adjacent in . The order and size of the jump graph are and , respectively. Hence, the average degree of is
Finally, we consider the semitotal line graph . Its vertex set is where two vertices are adjacent in if and only if they are either adjacent or incident in . Also, and , implying that
So, it is proved.
Theorem 5. Let be a graph of order and size . Then, the average degree of the line graph , total graph , jump graph , and semitotal line graph is as follows:
6. Average Degree of Some Graph Operations
Let and be two graphs such that and . In this final section, we study the average degree of some graph operations, namely, union, join, and corona products of two graphs and .
The union of two graphs and having disjoint vertex and edge sets is obtained easily by taking union of vertex sets as its vertex set and the union of edge sets as the edge set.
Theorem 6. The average degree of the union of two graphs and is
Proof. By the definition of union operation, we can write that and . This proves the result.
For two graphs and with orders and and sizes and , respectively, the join operation of two graphs and having disjoint vertex sets and and edge sets and is the graph union together with all the edges between and .
Theorem 7. The average degree of the join of two graphs and is
Proof. By the definition, we can deduce that and . Hence, the result follows.
The corona product of two graphs and is defined to be the graph obtained by taking one copy of (which has vertices) and copies of , and then, joining the th vertex of to every vertex in the th copy of , for , we obtain the following.
Theorem 8. The average degree of the corona product of two graphs and is
Proof. We know that and . This gives the required number.
Data Availability
No data available.
Conflicts of Interest
The authors declare that they have no conflicts of interest.