Some new spectral bounds for graph irregularity

Abstract

The irregularity of a simple graph $G=(V,E)$ is defined as

$$\text{irr}\left(G\right)=\sum _{uv\in E\left(G\right)}|d_G\left(u\right)-d_G\left(v\right)|,$$

where d_G(u) denotes the degree of a vertex u ∈ V(G). This graph invariant, introduced by Albertson in 1997, is a measure of the defect of regularity of a graph. Recently, it also gains interest in Chemical Graph Theory, where it is named the third Zagreb index. In this paper, by means of the Laplacian eigenvalues and the normalized Laplacian eigenvalues of G, we establish some new spectral upper bounds for irr(G). We then compare these new bounds with a known bound by Goldberg, and it turns out that our bounds are better than the Goldberg bound in most cases. We also present two spectral lower bounds on irr(G).

Introduction

All graphs throughout this paper are finite, undirected and simple. Let G be such a graph with vertex set $V\left(G\right)=\{v_1,v_2,\dots ,v_n\}$ and edge set E(G). Denote by d_G(v_i) (or simply, d_i) the degree of vertex v_i in G, $i=1,2,\dots ,n$. Let $(d_1,d_2,\dots ,d_n)$ be the degree sequence of G. We also write Δ(G) and δ(G) for the maximum vertex degree and the minimum vertex degree of G, respectively. A graph G is said to be regular if $\Delta \left(G\right)=\delta \left(G\right)$.

The adjacency matrix of a graph G is $A\left(G\right)={\left(a_{ij}\right)}_{n\times n},$ where elements $a_{ij}=1$ if the vertices v_i and v_j in G are adjacent, and $a_{ij}=0$ otherwise. The Laplacian matrix of G is defined to be $L\left(G\right)=D\left(G\right)-A\left(G\right),$ where D(G) is the diagonal matrix of vertex degrees of G. If G has no isolated vertices, then the normalized version of L(G) is defined as $\mathcal{L}\left(G\right)=D{\left(G\right)}^{-1/2}L\left(G\right)D{\left(G\right)}^{-1/2}$. It is well known that both L(G) and $\mathcal{L}\left(G\right)$ are positive semi-definite matrices, and hence their eigenvalues (always known as the Laplacian eigenvalues and the normalized Laplacian eigenvalues of G, respectively) are nonnegative real numbers, which could be ordered by μ₁(G) ≥ μ₂(G) ≥ ⋅⋅⋅ ≥ μ_n(G) and ν₁(G) ≥ ν₂(G) ≥ ⋅⋅⋅ ≥ ν_n(G), respectively. For more details regarding the Laplacian eigenvalues and the normalized Laplacian eigenvalues of a graph, one may refer to [5], [6].

In 1997, Albertson [2] defined the imbalance of an edge $e=uv\in E\left(G\right)$ as $|d_G\left(u\right)-d_G\left(v\right)|,$ and the irregularity of a graph G as

$$\begin{array}{c}\hfill \text{irr}\left(G\right)=\sum _{uv\in E\left(G\right)}|d_G\left(u\right)-d_G\left(v\right)|.\end{array}$$

Clearly, for a connected graph G, $\text{irr}\left(G\right)=0$ if and only if G is regular, and for a non-regular graph G, irr(G) is a measure of the defect of regularity of G. In [2], Albertson first proved the following upper bound on irr(G):

$$\begin{array}{c}\hfill \text{irr}\left(G\right)\le \frac{4n^3}{27},\end{array}$$

which was later improved by Abdo et al. [1] as

$$\begin{array}{c}\hfill \text{irr}\left(G\right)\le \lfloor \frac{n}{3}\rfloor \lceil \frac{2n}{3}\rceil (\lceil \frac{2n}{3}\rceil -1).\end{array}$$

On the other hand, Zhou and Luo [17] showed that if G is a graph with n vertices and m edges, then

$$\begin{array}{c}\hfill \text{irr}\left(G\right)\le \sqrt{m(n{M}_1\left(G\right)-4m^2)},\end{array}$$

where $M_1\left(G\right)={\sum }_{v\in V\left(G\right)}{d}_G{\left(v\right)}^2,$ which is referred to as the first Zagreb index. Remark that this index and the second Zagreb index, defined by $M_2\left(G\right)={\sum }_{uv\in E\left(G\right)}{d}_G\left(u\right)d_G\left(v\right),$ are two of the oldest and most investigated topological graph indices in Chemical Graph Theory, for details about the mathematical theory and the chemical applications of the Zagreb indices one can see [7], [8], [10], [14], [15], [16] and the references cited therein. It is also worth mentioning that in [8] Fath–Tabar established several obvious connections between the sum in (1.1) and the (first and second) Zagreb indices, and based on these connections he named the sum in (1.1) the third Zagreb index and denoted it by M₃(G). However, in the rest of the paper, we shall use its older name, i.e., the irregularity of a graph G, and denote it by irr(G).

There also have been some other upper bounds on irr(G) obtained by several authors such as Hansen and Mélot [11], Henning and Rautenbach [12], and Fath–Tabar [8]. Strictly speaking, these bounds, together with the Zhou–Luo bound (1.2), are noncomparable, but the Zhou-Luo bound (1.2) seems to be much sharper than the others for most graphs. Recently, by utilizing the spectral techniques, Goldberg made a further improvement upon the Zhou–Luo bound (1.2).

Theorem 1.1

(Goldberg [9]) If G is a graph on n vertices and with m edges, then

$$\begin{array}{c}\hfill \text{irr}\left(G\right)\le \sqrt{m(n{M}_1\left(G\right)-4m^2)({\mu }_1\left(G\right)/n)}.\end{array}$$

In this paper, by means of the Laplacian eigenvalues and the normalized Laplacian eigenvalues of G, we establish some new spectral upper bounds for irr(G). We then compare these new bounds with the Goldberg bound (1.3), and it turns out that our bounds are better in most cases. We also give two spectral lower bounds on irr(G).