On the Laplacian Coefficients of Bicyclic Graphs

In this paper, we investigate how the Laplacian coefficients changed after some graph transformations. So, I express some results about Laplacian coefficients ordering of graphs, focusing our attention to the bicyclic graphs. Finally, as an application of these results, we discuss the ordering of graphs based on their Laplacian like energy.


Introduction
τ is the number of spanning trees of G (see [1]).
Recently, the study on the Laplacian coefficient attracts much attention. Some works on Laplacian coefficients can be found in [2][3][4][5][6][7][8]. In this paper, we determine the largest coefficient among all the bicyclic graphs of order n .
Bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one. Let n B be the set of all connected bicyclic graphs of order n .

Transformations and Lemmas
Let F be a spanning forest of G with components , 1, 2, , If v is a pendant vertex of the graph G , adjacent to the vertex u , then the matching numbers of G conform to the recurrence relation

Lemma 2.12
Let G and G γ be the unicyclic graphs as shown in Fig. 2, l , r and s are positive integers.
Suppose that 1 r s ≤ ≤ . Then for every 0,1, , , Let k F and k F γ be the sets of spanning forests of G and G γ with exactly k components, respectively. We distinguish the following two cases. Case 1.
1 l = . According to the definition of the spanning forest, the all edges in the cycle p C can't exist in any forest F at the same time, where F is the arbitrary forests in k F . We distinguish the following two cases.
Case 1.1. If we remove the edge 1 1 u v in G and remove the edge 1 r u w in G γ , we can get the trees T and T γ . It is easy to see that T T γ = .

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On the Laplacian Coefficients of Bicyclic Graphs respectively. According to Definition 2.6, we know that At the same time, we can get the number of trees obtained in G γ more than that of trees obtained in G . By Lemma 2.1, it is easy to see that for Fig. 3), it is easy to see that the relationship between G γ ′ and G γ is similar to the relationship between G and G γ in Case 1.
According to the proof of Case 1, we have that for every 0,1, , ,

Laplacian Coefficients of Bicyclic Graphs
Definition 3.1 Let w be a vertex of the cycle p C (or q C , l p C + , p q C + ) in a connected bicyclic graph G .
Assume that G has a pendant path is a graph obtained by removing edge 1 ua and adding edge uv . We say that 2 G is a 2 η -transformation} of G (see Fig. 4, if 1 l = , see Fig. 5).

Laplacian Coefficients of Bicyclic Graphs
Definition 3.1 Let w be a vertex of the cycle p C (or q C , l p C + , p q C + ) in a connected bicyclic graph G .
Assume that G has a pendant path is a graph obtained by removing edge 1 ua and adding edge uv . We say that 2 G is a 2 η -transformation} of G (see Fig. 4, if 1 l = , see Fig. 5).     1 l > . The next proof is the similar to that of Case 1. Then the conclusion of Theorem 3.5 holds.
According to the Theorem 3.5, we know that the graph with maximum Laplacian coefficients must be obtained from ( , , ) l p q Ρ We say that 3 G is a 3 η -transformation of G (see Fig.   6).