New Measures of Graph Irregularity

In this paper, we define and compare three new measures of graph irregularity. We use these measures to tighten upper bounds for the chromatic number and the Colin de Verdiere parameter. We also strengthen the concise Turan theorem for irregular graphs and investigate to what extent Turan's theorem can be similarly strengthened for generalized r-partite graphs. We conclude by relating these new measures to the Randic index and using the measures to devise new normalised indices of network heterogeneity.


Introduction
Let G be a simple and undirected graph with vertex set V with |V | = n, m edges, t triangles, clique number ω and vertex degrees d i , for i = 1, . . . , n. Let µ 1 denote the largest eigenvalue of the adjacency matrix of G and let d denote the average degree.
We also use a parameter c v , introduced by Edwards [8], which he termed the "vertex degree coefficient of variation" and defined as follows: Edwards [8] showed that c v = 0 if and only if a graph is regular, so ν ≥ 1, with equality only for regular graphs.

Turán's Theorem for irregular graphs
Turán's Theorem, proved in 1941, is a fundamental result in extremal graph theory. In its concise form it states that: We strengthen this bound for irregular graphs as follows, and provide spectral and non-spectral proofs.
Before presenting the proofs we explain briefly the intuition underlying the above inequality. Theorem 1 is unusual because it involves m on both sides. A useful way to interpret the theorem is that ν is a measure of graph irregularity. Therefore all graphs with a given clique number and, for example, irregularity as measured by ν ≥ 2 have a maximum number of edges that is at most half of the number implied by Turán's Theorem.
Proof 1 (spectral). Nikiforov [12] has used the Motzkin-Straus inequality to prove a conjecture due to Edwards and Elphick [9] that: Hofmeister [10] proved that µ 2 1 ≥ n i=1 d 2 i /n. Therefore: Therefore: Proof 2 (non-spectral). This proof of Theorem 1 is based on a 1962 proof of the concise Turán Theorem due to Moon and Moser [11], as written up in an award winning paper by Martin Aigner entitled "Turán's Graph Theorem". Let C h denote the set of h-cliques in G with |C h | = c h . So for example, c 1 = n, c 2 = m, c 3 = t etc. For A ∈ C h let d(A) equal the number of (h + 1) cliques containing A. Moon and Moser [11] proved that: They also proved that: so with h = 2 this becomes: which is equivalent to: This definition of θ differs from that in [11] and enables the strengthening of Moon and Moser's proof. Combining (2) and (3) we have: To prove Theorem 1 we need to show that θ ≤ k − 1 for graphs without k-cliques. Consider the claim: For h = 2, this is inequality (4). We therefore use induction on h and (1) as follows: as claimed in (5). Now if G contains no k-clique then c k = 0 and we infer θ ≤ h = k − 1 from (5).
Theorem 1 is exact for star graphs. The full form of Turán's theorem states that m(G) ≤ m(T r (n)), where T r (n) is the complete r-partite graph of order n whose classes differ by at most one, with equality holding only if G = T r (n). It is not the case that for all irregular graphs m(G) ≤ m(T r (n))/ν.

Generalized r-partite graphs
In a series of papers, Bojilov and others have generalized the concept of an r-partite graph. They define the parameter φ to be the smallest integer r for which V (G) has an r-partition: for all v ∈ V i and for i = 1, 2, . . . , r.
It is notable that φ depends only on the degrees of G, and not on the adjacency matrix of G. Indeed, φ is defined for any set of n integers a i , where 0 ≤ a i ≤ n − 1, which may or may not correspond to the degrees of a graph.
Theorem 2.1 in [3] proves that φ is a lower bound for the clique number and the greedy Algorithm 1 and Theorem 3.1 in [3] demonstrate that φ can be computed in linear time. For d-regular graphs, Theorem 4.4 in [3] proves that: Khadzhiivanov and Nenov [7] have proved that φ satisfies Turán's Theorem: Theorem 4.1 in [3] provides a simpler proof of (6). The study of φ has therefore led to a novel proof of the concise version of Turán's Theorem, which also demonstrates that this famous result is in fact a function only of the degrees of a graph rather than its adjacency matrix.
Bojilov and Nenov [2] have strengthened (6) as follows: This result is further strengthened in Theorem 5.4 in [3] where it is shown that: (7) is equivalent to r = 2 in this chain of inequalities.
It is therefore natural to ask whether φ can replace ω in Theorem 1? The answer is no, because, for example, the graph in Figure 1 provides a counter-example. There are also various spectral lower bounds for ω of which the simplest, due to Cvetkovic [6], is: The graph in Figure 2 is an example of a graph which does not satisfy (8), with ω replaced by φ. It also demonstrates that a variety of other spectral lower bounds for ω are not lower bounds for φ. Furthermore, φ does not satisfy the Motzkin-Straus inequality. Theorem 1, when expressed as a lower bound for ω, exceeds the best of the lower bounds for φ above, for many but not all graphs.
We have established that φ satisfies (7) but does not satisfy Theorem 1. This begs the question as to whether there is a graph parameter which satifies Turán's Theorem but does not satisfy (7)? The answer is yes. Following the notation in [3], letḠ denote the complement of G and let: where CW denotes the Caro-Wei lower bound for the independence number. Corollary 6.3 in [3] demonstrates that CW (Ḡ) ≤ φ(G) and that CW (Ḡ) satisfies Turán's Theorem. The graph in Figure 3 on 7 vertices with degree sequence (3, 3, 2, 2, 2, 1, 1) has CW (Ḡ) = 1.43, φ = 2 and ω = 3 and does not satisfy (7), with φ replaced by CW (Ḡ).  [3] demonstrates that φ β ≤ φ and Corollary 6.3 demonstrates that φ β satisfies Turán's Theorem. The authors note, however, that they do not know if φ β (G) ≤ CW (Ḡ) for all graphs. Consider the graph in Figure 4 which has 8 vertices and degree sequence (7,5,4,4,4,4,3,3). This graph has φ = 3 and CW (Ḡ) = 2.73 but φ β is an integer and it is not hard to demonstrate that φ β = 2, because with these degrees there cannot be a subset of size 5 or more and neither can there be two classes of size 4. It is therefore not the case that φ β (G) ≤ CW (Ḡ) for all graphs.

Conclusion
As mentioned in the Introduction, c v is a measure of the irregularity of a graph. It is therefore of interest to compare it with other measures of irregularity. Collatz and Sinogowitz [5] proposed a spectral measure, namely ε(G) = µ 1 − d.
Bell [1] proposed a variance measure, namely var(G) = (d i − d) 2 /n = (d 2 i /n) − d 2 and identified the most irregular graphs for both measures. He also showed that the measures are incompatible for some pairs of graphs.
The earlier measure due to Edwards [8], c v , is closely related to var(G) since: var(G) = (c v d) 2 .
Nikiforov [13] has proved that: var(G) 2 √ 2m ≤ ε(G) ≤ 4 n 2 var(G) Consequently: Given the role of c v in Theorem 1 it maybe that the use of c v as a measure of irregularity merits further investigation.