The general Albertson irregularity index of graphs

We introduce the general Albertson irregularity index of a connected graph $G$ and define it as $A_{p}(G) =(\sum_{uv\in E(G)}|d(u)-d(v)|^p)^{\frac{1}{p}}$, where $p$ is a positive real number and $d(v)$ is the degree of the vertex $v$ in $G$. The new index is not only generalization of the well-known Albertson irregularity index and $\sigma$-index, but also it is the Minkowski norm of the degree of vertex. We present lower and upper bounds on the general Albertson irregularity index. In addition, we study the extremal value on the general Albertson irregularity index for trees of given order. Finally, we give the calculation formula of the general Albertson index of generalized Bethe trees and Kragujevac trees.


Introduction
Let G be a simple undirected connected graph with vertex set V (G) and edge set E(G). uv ∈ E(G). Denote by P n and K 1, n−1 the path and star with n vertices, respectively.
In 1997, the Albertson irregularity index of a connected graph G, introduced by Albertson [1], is defined as This index has been of interest to mathematicians, chemists and scientists from related fields due to the fact that the Albertson irregularity index plays a major role in irregularity measures of graphs [3,4,7,8,17], predicting the biological activities and properties of chemical compounds in the QSAR/QSPR modeling [12,24] and the quantitative characterization of network heterogeneity [9]. By the natural extension of the Albertson irregularity index, Gutman et al. [13] recently proposed the σ-index as follows: where F and M 2 are well-known the forgotten topological index and the second Zagreb index of a graph G, respectively. Recently, the σ-index of a connected graph G is studied, such as the characterization of extremal graphs [5] and mathematical relations between the σ-index and other graph irregularity indices [21].
The generalization of topological index is a trend of mathematical chemistry in recent years. Many classical topological indices are generalized, such as the general Randić index [6], the first general Zagreb index [18], the general sum-connectivity index [31], the general eccentric connectivity index [28], etc. Motivated by this fact, we propose the general Albertson irregularity index of a graph G as follows: where p is a positive real number. Evidently, A 1 (G) = Alb(G) and A 2 2 (G) = σ(G). The other motivation is that the topological index formed from distance function of the degree of vertex has attracted extensive attention of scholars. In 2021, Gutman [10] proposed the Sombor index of a graph G and defined it as SO(G) = uv∈E(G) d 2 (u) + d 2 (v), which is the Euclidean norm of d(u) and d(v). According to Gutman [11], it is imaginable to use other distance function to study properties of graphs. Based on this, it is not difficult to find that A p (G) is the Minkowski norm of d(u) and d(v), which is unification of absolute distance, Euclidean distance and Chebyshev distance. Hence A p (G) = ∆−δ as p becomes infinite. In particular, A p (G) is the l p -norm of d(u) and d(v) for p ≥ 1.
We will first recall some useful notions and lemmas used further in Section 2. In Section 3, upper and lower bounds on the general Albertson irregularity index of graphs are given, and the extremal graphs are characterized. In Section 4, the first two trees with minimum general Albertson irregularity index are determined in all trees of fixed order.
In Section 5, the general Albertson index of the well-known generalized Bethe trees and Kragujevac trees is obtained.

Preliminaries
Let u ∨ G be the graph by adding all edges between the vertex u and V (G). The first general Zagreb index of a graph G is defined as Z p (G) = v∈V (G) d p (v) for any real number p. The distance between two vertices u, v ∈ V (G), denoted by d(u, v), is defined as the length of a shortest path between u and v. The eccentricity of v, ε(v), is the distance between v and any vertex which is furthest from v in G. The line graph L(G) is the graph whose vertex set are the edges in G, where two vertices are adjacent if the corresponding edges in G have a common vertex. Let T n be the set of trees with n vertices. A spider is a tree with at most one vertex of degree more than two.
with equality if and only if x 1 = x 2 = · · · = x n .
Lemma 2.2. (Hölder inequality) Let (a 1 , a 2 , . . . , a n ) and (b 1 , b 2 , . . . , b n ) be two n-tuples of real numbers and let p, q be two positive real numbers such that 1 with equality if and only if |a i | p = λ|b i | q for some real constant λ, 1 ≤ i ≤ n.
. .) be a non-zero vector. Then for p ≥ 2, with equality if and only if all but one of the x i are equal to 0.
If p > 0 and ∆ = d(u) = 3, then we have Combining the above arguments, we have the proof.

Some bounds for the general Albertson index
Theorem 3.1. Let G be a connected graph with m edges. If p > q, then with equality if and only if G is a regular graph (when G is non-bipartite) or G is a (∆, δ)-semiregular bipartite graph (when G is bipartite).
Proof. By Lemma 2.1, we have that is, with equality if and only if G is a regular graph (when G is non-bipartite) or G is a (∆, δ)-semiregular bipartite graph (when G is bipartite).
Theorem 3.3. Let G be a connected graph. If 1 p + 1 q = 1, then with equality if and only if p = 2, or G is a regular graph (when G is non-bipartite) or G is a (∆, δ)-semiregular bipartite graph (when G is bipartite).
with equality if and only if p = 2, or G is a regular graph (when G is non-bipartite) or G is a (∆, δ)-semiregular bipartite graph (when G is bipartite).
Theorem 3.4. Let G be a connected graph with m edges. If p ≥ 1 is an integer, then that is, Theorem 3.5. Let G be a connected graph with m edges. If p ≥ 1, then for p ≥ 1. By Bernoulli inequality, we have that is, For 0 < p < 1, by Lemma 2.4 and Bernoulli inequality, we have the proof.  Proof. By definition of A p (G), we have

Proof. By Lemma 2.5, we have
with equality if and only if (d(u) − 1)(d(v) − 1) ≤ 0, that is, d(v) = 1 for every edge uv in G, that is, G is a star K 1, n−1 .
Corollary 3.9. Let G be a connected graph with n vertices and m edges. Then with equality if and only if G ∼ = K 1, n−1 .
Theorem 3.10. Let G be a connected graph with n vertices. Then where G is the complement of G.
Proof. By definition of u ∨ G, we have Corollary 3.11. Let G be a connected graph with n vertices and m edges. Then Theorem 3.12. Let u be a pendant vertex of a connected graph G with n ≥ 3 vertices.
If G + P t (t ≥ 1) is the graph by adding a new (pendant) path to u, then (ii) A p (G + P t ) = A p (G) for p = 1.
(iii) A p (G + P t ) < A p (G) for p > 1.

Proof.
Let v be the unique neighbour of u in G. Since a p + b p > (a + b) p for a > 0, b > 0 and 0 < p < 1, we have . By a similar reasoning as above, we have the proof of (ii) and (iii).
Corollary 3.13. Let u be a pendant vertex of a connected graph G with n ≥ 3 vertices.
If G + P t (t ≥ 1) is the graph by adding a new (pendant) path to u, then σ(G + P t ) < σ(G).

The general Albertson index of trees
Theorem 4.1. Let T n ∈ T n . Then The lower bound is attained if and only if T n ∼ = P n . The upper bound is attained if and only if T n ∼ = K 1, n−1 .
Proof. If ∆ ≥ 3, then T n has at least three pendant vertices. Thus A p (T n ) > 3  (ii) If p = 1, then A p (T n ) ≥ 6 with equality if and only if T n is a spider with ∆ = 3.
Proof. Let d 1 ≥ d 2 ≥ d 3 ≥ · · · ≥ d n be the degree sequence of T n , and let k be the number of non-pendant edges uv with d(u) = d(v). Then Theorem 4.3. Let T n be a tree with n vertices. If p ≥ 1, then where v ∆ is a vertex of maximum degree.
is a increasing and convex function for x > 0 and p ≥ 1, we have Since T n has at least ∆ pendant vertices, we have where l 1 , l 2 , . . . , l ∆ is the distance from maximum degree vertex v ∆ to pendant vertex v l i , Corollary 4.4. Let T n be a tree with n vertices. Then with equality if and only if G is a spider.

Kragujevac trees
In this section, we give the calculation formula of the general Albertson index of generalized Bethe trees and Kragujevac trees which are a wide range of applications in the field of mathematics [22,25], Cheminformatics [15,27,29], statistical mechanics [19], etc.
A generalized Bethe tree [23] is a rooted tree in which vertices of the same level (height) have the same degree. We usually use B k to denote the generalized Bethe tree with k levels with the root at the level 1. More specifically, B k, d denotes a Bethe tree [16] of k levels with root degree d, and the vertices between level 2 and k − 1 all have degree d + 1.
A regular dendrimer tree [14] T k, d is a special case of B k , where the degrees of all internal vertices are d.
Proof. By definition of the generalized Bethe tree, we have Thus we have the proof. Then (2k i + 1) = n, by definition of the Kragujevac tree, we have Thus we have the proof.

Conclusion
In this paper, we propose the general Albertson irregularity index which extends classical interaction among A p (G), irr p (G) and N p (G) will be carried out in the near future.