Abstract

A simple graph is said to be regular if its vertices have the same number of neighbors. Otherwise, is nonregular. So far, various formulas, such as the Albertson index, total Albertson index, and degree deviation, have been introduced to quantify the irregularity of a graph. In this paper, we present sharp lower bounds for these indices in terms of the order, size, maximum degree, minimum degree, and forgotten and Zagreb indices of the underlying graph. We also prove that if has the minimum value of degree deviation, among all nonregular -graphs, then .

1. Introduction

Let be a graph and let be a vertex of . We say that is an -graph if its order and size are and , respectively. As usual, we use to denote the degree of the vertex . Moreover, and are used to denote the maximum and minimum vertex degrees, respectively.

In [1], the first Zagreb and the second Zagreb indices were introduced as follows:

The forgotten topological index, , is defined as follows [2]:

A graph is said to be -regular if . If , then is called -regular or regular for brevity; otherwise, is nonregular. Until now, various formulas have been introduced to quantify the irregularity of a graph. We refer to the interesting readers to consult the recently published book of Ali et al. [3] for more information on this topic. Albertson in [4] introduced the following formula as an irregularity measure of a connected graph :

Today, this parameter is known as the Albertson index and is denoted by . A few years later, Nikiforov presented the degree deviation as follows [5]:

Moreover, Abdo et al. in [6] expanded the Albertson index as follows:

They called it as the total irregularity index and denoted it by . We encourage the interested reader to study references [7–16] for more details on these irregularity measures. Although some graphs with equal irregularity indices are presented in [17], there are some comparisons between these parameters in [18], which may imply the unreliability of them. In [19], relation between the chromatic number and irregularity measures was investigated. In [20–22], some bounds on of graphs were given. In the next section, we obtain further sharp lower bounds for (total) Albertson indices and degree deviation of a graph in terms of its order, size, maximum degree, minimum degree, forgotten, and Zagreb indices. Finally, we show that if has the minimum value of among all nonregular -graphs, then .

2. Main Results

In this section, we give lower bounds for (total) Albertson indices and the degree deviation of a graph . We end this section with a result related to the minimality of .

Theorem 1. Let be a nonregular -graph. Then,and the equality holds if and only if is -regular.

Proof. For each ,and the equality holds if and only if . Then,and the equality holds if and only if , and the this relation holds if and only if is -regular. On the other hand,Therefore, by relation (8), we conclude thatand the equality holds if and only if is -regular.

Corollary 1. If is a nonregular -graph, thenand the equality holds if and only if is -regular. Also,and the equality holds if and only if is -regular.

Proof. For each , . This yields that for each , we have(i) and the equality holds if and only if (ii) and the equality holds if and only if .From (i), we getand the equality holds if and only if is -regular. Also, according to (ii), we can writeand the equality holds if and only if is -regular. Due to the abovementioned argument, we can easily conclude thatand the equality holds if and only if is -regular. We can also easily drive thatand the equality holds if and only if G is -regular. Using Theorem 1,and the equality holds if and only if is -regular. Also,and the equality holds if and only if is -regular.

Theorem 2. Let be a connected nonregular -graph. Then,and the equality holds if and only if is -regular.

Proof. For each , and the equality holds if and only if . This implies thatand the equality holds if and only if . Since is connected, then we can say that the equality holds if and only if is -regular. On the other hand,Then, using equation (20), we haveand the equality holds if and only if is -regular.

Theorem 3. If is a nonregular -graph, thenand the equality holds if and only if is -regular or -regular.

Proof. Let . Let also that for . Then, it is evident thatThen, the relation follows:and the equality holds if and only if and the same sign. On the other hand, for , we haveand the left equality holds if and only if and the right equality holds if and only if . Thus,and the left equality holds if and only if and the right equality holds if and only if . Consequently, and the equality holds if and only if . Then,and the equality holds if and only if . Thus, by relation (25), we arrive at the relation:Moreover, if and only if the equalities hold in relations (20) and (28) for , and these are satisfied if and only if is -regular or -regular. On the other hand,By using relation (30), and the equality holds if and only if is -regular or -regular.
By a similar argument applied in Corollary 1, we can obtain the next result.

Corollary 2. Let be a nonregular -graph. Then,and the equality holds if and only if is -regular. Also,and the equality holds if and only if is -regular.

Lemma 1. Let be an -graph. If , then at most one of the relations and holds.

Proof. Assume, to the contrary, that the both of relations and hold. Thus, , which leads to a contradiction because .

Theorem 4. Let be an -graph. If , then there exists an -graph such that , , and .

Proof. Let and be two vertices of such that , and . Then, , and consequently, we suppose and define . Thus, is an -graph such that , , and . Then, . Now, we consider three cases by Lemma 1 for computing :(i) and . In this case, by the definition of S, we have(ii) and . In this case, we have(iii) and . Then,Using cases (i–iii) and Lemma 1, we reach to , which completes the proof.

Corollary 3. If has the minimum value of among all nonregular -graphs, then .

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.