The Matching Energy of Random Multipartite Graphs

Let Gn be a simple graph with n vertices. Gutman and Wagner founded the theory of random graphs, they introduced the matching energy of the graph Gn, which was defined as the sum of the absolute values of the eigenvalues of the matching polynomial of the graph Gn. For the Erdös-Rényi type random graph Gn,p of order n with a fixed probability p, where p is a real number greater than zero and less than 1, that is, the graph G on n vertices by connecting two vertices with probability p(e), and each edge is independent of other one. Chen, Li and Lian solved a conjecture proposed by Gutman and Wagner, that is, the expectation of the matching energy of Gn,p converges to a certain number associated with n and p almost surely. But they only did the result for random bipartite graphs. In this paper, we give some lower bounds for the matching energy of random bipartite graphs. And then we will use Chen et al’s method to generalize this conclusion to any random multipartite graphs.


Introduction
Let G be a simple graph with vertex set V (G) and edge set E(G). And its vertex number and edge number are denoted by |V(G)| and |E(G)|, respectively. A matching of G is a set of independent edges in G, and a k-matching of G is a matching of G that has exactly k edges, denoted by ( , ) M G k . Let m(G, k) be the number of k-matchings, i.e., the number of selections of k independent edges, or the number of k-element independent edge sets of a graph G. Specifically, The matching polynomial has many important implications in statistical physics and chemistry; see [6,9,14]. Many results on the properties of the matching roots have been obtained; see [4,[10][11][12][13].
On the other hand, the matching energy of a graph G was introduced by Gutman and Wagner [5], which is defined as the sum of the absolute values of the roots of the matching polynomial of G (see Eq. (3)). Moreover, Gutman and Wagner found that the matching energy is a quantity of relevance with chemical applications [5] and they arrived at the relation where TRE(G) is the topological resonance energy of the graph G, and E(G) is the energy of the graph G, i.e., the sum of the absolute values of the roots of the matching polynomial of G. For more information about the applications of matching energy, we refer to [6,9]. For terminology and notation not given here, we refer to [1].
Defifinition 1 [5]. Let G be a simple graph, and let 1 2 , , , n µ µ µ ⋯ be the zeros of its matching polynomial. Then the matching energy of G is defifined as Theorem 2 [5]. Let G be a simple graph. The matching energy of G can be expressed as the following formula Then Eq. (3) could be considered as the definition of matching energy and Eq. (2) would become a theorem.
Next the basic properties of the matching energy and the graphs with extremal matching energy will be introduced.
Theorem 3 [8]. If the graph G is a forest, then its matching energy coincides with its energy.
Theorem 4 [8]. Let G be a graph and e is one of its edges. Let G-e be the subgraph obtained by deleting from G the edge e, but keeping all the vertices of G. Then ( ) ( ). ME G e ME G − < Theorem 5 [8]. Among all graphs on n vertices, the empty graph without edges and the complete graph n K have, respectively, minimum and maximum matching energy.
By theorem 4, the connected graph with minimal ME must be a tree. By theorem 3, trees have equal E-and ME-values. The fact that, among n-vertex trees, the star has minimal energy was established long time ago [6]. Therefore, we have Theorem 6 [8]. The connected graph on n vertices having minimum matching energy is the star n S . The matching energy of the empty graph is clearly 0, and the matching energy of the star, which equals its ordinary energy, is ( ) 2 ( 1). n ME S n = − There does not seem to be a similarly simple expression for the maximum matching energy ( ) n ME K , but the following could be proven: Theorem 7 [5]. The matching energy of the complete graph n K is asymptotically equal to 3 2 8 . 3 n π . More precisely, 3 We can directly extend Proposition 1 into: Theorem 11 [5]. The bipartite graph on n vertices having maximum matching energy is In the 1980s the present author established a number of relations of the type G H > [15][16][17][18][19]. Lemma 5 (Sliding). Let G be a connected graph with at least two vertices, and let u be one of its vertices. Denote by P(n,k,G,u) the graph obtained by identifying u with the vertex k v of a simple path 1 2 , ,..., . 2 1, , )) ME P n G u ME p n G u ME p n p l G u ME P n p G u Lemma 6 (Ironing). Suppose that G is a connected graph and T is an induced subgraph of G such that T is a tree and T is connected to the rest of G only by a cut vertex v. If T is replaced by a star of the same order, centered at v, then the matching energy decreases (unless T is already such a star). If T is replaced by a path, with one end at v, then the matching energy increases (unless T is already such a path).
Based on Lemmas 5 and 6, it was possible to characterize unicyclic, bicyclic, and tricyclic connected graphs with smallest and greatest matching energy. The consideration of trees will be skipped since, by theorem 3 it reduces to the well known case of extremal energy.
Denote by n U the set of all connected unicyclic graphs on n vertices. Let n C be the n-vertex cycle, and let n S + be the graph obtained by adding a new edge to the star n S . of course, , .
n n n C S U + ∈ ( ) ( ) ( ) ( ). ME S ME G ME TC ME TC • < < = In 1950s, Erd o ɺɺ s-R e′ nyi [3] founded the theory of random graphs. The Erd o ɺɺ s-R e′ nyi random graph model ( ) n G p consists of all graphs on n vertices in which the edges are chosen independently with probability p, where p is a constant with 0 < p < 1.
Chen, Li and Lian in [2] solved a conjecture proposed by Gutman and Wagner, which is now a result stated as follows.
Theorem 15 [2]. For (0,1) p ∈ , the matching energy , ( ) n p ME G of the random graph , n p G enjoys asymptotically almost surely the following equation: In this paper, we give some lower bounds for the matching energy of random bipartite graphs. And then we will use Chen et al's method to generalize this conclusion to any random multipartite graphs, and obtain some lower bounds for the matching energy of random multipartite graphs.

The Matching Energy of Random Bipartite Graphs
In this section, we will give a lower bound for the matching energy of random bipartite graphs.
Theorem 16. Let 1 G and 2 G be two graphs with the same number of vertices (see Figure 1), and let 1 G be connected and bipartite with 1 ( ) V G A B = ∪ and 2 G be disconnected with two connected components 1 2 G and 2 2 G , i.e., For any nonnegative integer k, we claim that We distinguish the following cases. In other words, any set of k-matchings of G is also a one of

The Matching Energy of Random Multipartite Graphs
Next, we will give a lower bound for the matching energy of random multipartite graphs. Theorem 18. Let 1 G and 2 G be two graphs with the same number of vertices (see Figure 2), and let 1 G be connected and k-partite with