Algebraic Connectivity and Disjoint Vertex Subsets of Graphs

A graph G is often used to model a complex network. +e vertex set and the edge set of graph G are denoted byN and E, respectively. A network is represented as an undirected graph G � (N,E) consisting of N � |N| nodes and E � |E| links, respectively. Graph theory has provided chemists with a variety of useful tools, such as in the topological structure [1–3]. +e Laplacian matrix of a graph G is denoted by L, and L � D − A, where D is a diagonal matrix whose diagonal entries are its degrees and A is the adjacencymatrix of G.+e Laplacian eigenvalues of a graph G are the eigenvalues of L, denoted by 0 � μN ≤ μN− 1 ≤ . . . ≤ μ1, which are all real and nonnegative. +e second smallest Laplacian eigenvalue μN− 1 of a graph is well known as the algebraic connectivity, which was first studied by Fiedler [4].+e algebraic connectivity [5] of a graph is important for the connectivity of a graph [6], which can be used to measure the robustness of a graph. It has been emerged as an important parameter in many system problems [7–18]. Especially, the algebraic connectivity also plays an important role in the partitions of a complex network. For the literature on the algebraic connectivity of a graph [19], the reader is referred to [20, 21]. In this work, the relationships are researched between the algebraic connectivity and disjoint vertex subsets of graphs, which are presented through some upper bounds. 2. Preliminaries


Introduction
A graph G is often used to model a complex network. e vertex set and the edge set of graph G are denoted by N and E, respectively. A network is represented as an undirected graph G � (N, E) consisting of N � |N| nodes and E � |E| links, respectively.
Graph theory has provided chemists with a variety of useful tools, such as in the topological structure [1][2][3]. e Laplacian matrix of a graph G is denoted by L, and L � D − A, where D is a diagonal matrix whose diagonal entries are its degrees and A is the adjacency matrix of G. e Laplacian eigenvalues of a graph G are the eigenvalues of L, denoted by 0 � μ N ≤ μ N− 1 ≤ . . . ≤ μ 1 , which are all real and nonnegative. e second smallest Laplacian eigenvalue μ N− 1 of a graph is well known as the algebraic connectivity, which was first studied by Fiedler [4]. e algebraic connectivity [5] of a graph is important for the connectivity of a graph [6], which can be used to measure the robustness of a graph. It has been emerged as an important parameter in many system problems [7][8][9][10][11][12][13][14][15][16][17][18]. Especially, the algebraic connectivity also plays an important role in the partitions of a complex network. For the literature on the algebraic connectivity of a graph [19], the reader is referred to [20,21]. In this work, the relationships are researched between the algebraic connectivity and disjoint vertex subsets of graphs, which are presented through some upper bounds.

Preliminaries
Let x ∈ R n be a vector. Let B be an incidence matrix of G. en, For any vector x, y ∈ R n , the inner product of x and y is defined as (x, y). Two upper bounds on the algebraic connectivity are given as follows.
Lemma 1 (see [20]). For any vector f ∈ R n , the Rayleigh inequality is as follows: Lemma 2 (see [20]). For any vector f ∈ R n , we have Let A and B be two disjoint subsets of N, respectively. e distance between two disjoint subsets A and B of N is denoted by h (A, B). For continence, h takes the place of h(A, B). Let h(u, A) be the distance between the node u and A, which is the shortest distance of the node u ∈ N to a node of the set A. Suppose a � |A|/N and b � |B|/N. A result on the algebraic connectivity and two partitions of graphs is presented by Alon [22] and Milman [20] below. Lemma 3 (see [23]). For any two disjoint subsets A and B of N, it holds where E A and E B are the number of links in the sets A and B, respectively.
Moreover, the next step consider the case of three disjoint vertex subsets of graphs [24].

Main Result
Let A, B, and C be the subsets of N, respectively, where their numbers of nodes are, respectively, |A|, |B|, and |C|. Assume we construct a function g(u) related to node u as follows, where the constructed function is referred to the book [25]: . It is easy to check that (f, c) � 0, where c is a constant function [26][27][28][29][30]. Meanwhile, (g, c) ≠ 0 can be checked. us, the following cases need to discussed.

Case 1. If the node u belongs to any one subset of
By Case 1 and Case 2, (g, c) ≠ 0 holds. In contrast, if (g, c) � 0, then g(u) � 0 for each u ∈ N and g − g � 0, which is a contradiction with f � g − g ≠ 0. From the definition g(u), for any two adjacent nodes u and v, we have Our main result is as follows.
Proof. For subsets A, B, and C, by Lemma 2, we have From (2) and where g � 0, and since the coordinates of the center of gravity of the three regions are the average of the triangle region, then the vectors g(u, A) + g(u, B) + g(u, C) � 0. e sum of the vectors of the center of gravity of the triangle to the vertices is equal to 0 [25]. e center of gravity is analogous to the mean or average from statistics [6,31,32]: 2 Mathematical Problems in Engineering By the above inequalities and Lemma 2, it arrives that Example 1. Figure 1 describes the graphs G 1 and G 2 , each with N � 7 nodes, L � 10 links, and a diameter ρ � 4. For G 1 subsets, and h s � 0.5. eir algebraic connectivity [33] and their upper bounds on (11) are as follows. For the G 1 and G 2 aplacian matrixes, Figure 1: e graphs G 1 and G 2 .
For G 1 , the algebraic connectivity is 0.6338, and the algebraic connectivity of G 2 is 0.5858. For G 1 upper bounds A graph with the second smallest Laplacian eigenvalue μ N− 1 is thus more robust, in the sense of being better connected.
where g � 1 n in which g is the average of the A, B, C, D field.
Proof. For subsets A, B, and C, by Lemma 2, we have where links of the sets in the node sets A, B, C, and D are E A , E B , E C , and E D , respectively. From (2), we obtain