Kotlov、Lovász和Vempala[1]以graph labellings 定義了一個圖不變量ν並以此給出Colin de Verdière 圖不變量μ的另一個等價定義。當G ̸= \overline{K_2}時，這兩個圖不變量滿足等式μ(G)+ν(\overline{G})=|G|-1。這篇論文旨在探討μ和ν在一些為人熟知的圖運算下的行為，如Cartesian products、disjoint unions和graph joins。

首先我們引進“almost one-directional” labelling 來證明對於一系列的圖{G_i}的disjoint union，有不等式max ν(G_i)≤ ν(∪ G_i)≤ max ν(G_i)+1。而且我們提供了一個充分條件使得第一個等式成立。這近乎刻劃 在disjoint unions 下的行為。作為一個簡單的應用，我們能夠計算完全多部圖的μ值。除此之外，這個不等式也提供一些判斷不連通圖是否為ν-minimal graph 的必要條件。這也促使我們去探討，要如何用小的μ-maximal graphs 來生成一個給定的有separating cliques 的μ-maximal graph。藉由van der Holst、Lovász 和Schrijver 三人於[4]中所刻劃的μ在clique sum 下的行為，我們給出一個判斷μ-maximal graphs 的clique sum 是否仍是μ-maximal 的準則。最後我們證明ν在Cartesian products下的成長速率有一個以圖的片數為變數的線性成長的上界，而μ則有一個指數成長的下界。

Kotlov, Lovász and Vempala in [1] offered a reformulation for the Colin de Verdière graph invariant μ by introducing another graph invariant ν defined via graph labellings. These two parameters are related by the equality μ(G)+ν(\overline{G})=|G|-1 for G ̸= \overline{K_2}. In this paper we examine how these two invariants μ and ν vary under some well-known graph operations, such as Cartesian products, disjoint unions and graph joins.

First, we introduce "almost one-directional" labelling to derive that for the disjoint union of graphs {G_i}, max ν(G_i)≤ ν(∪ G_i)≤ max ν(G_i)+1. Also we show a sufficient condition for the first equality to hold. This nearly characterizes the behavior of ν under disjoint unions. As an application, we are able to compute the exact value μ for complete multipartite graphs. The inequality also provides us with some necessary conditions for a disconnected graph being ν-minimal. Therefore, this also motivates us to look into how μ-maximal graphs with separating cliques could be built up by smaller ones via clique sums. Using the characterization of μ under clique sum proved by van der Holst, Lovász and Schrijver in [4], we derive a criterion in judging whether a clique sum of two μ-maximal graphs is μ-maximal. Lastly, we show that the growth rate of ν under Cartesian products has a linear upper bound in the number of graphs while that of μ has a exponential lower bound in the number of graphs.

1 Introduction ...................................................1
2 Notations and terminology ......................................4
3 Characterizations of n under disjoint union ....................5
4 Decomposing m-maximal graphs with separating cliques ..........10
5 μ and ν under Cartesian products ..............................16

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