On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with 𝑛 Vertices and 𝑚 Pendent Vertices

Abstract

Let Φ(𝐺,𝜆)=det(𝜆𝐼_𝑛−𝐿(𝐺))=∑^𝑛_𝑘=0(−1)^𝑘𝑐_𝑘(𝐺)𝜆^𝑛−𝑘 be the characteristic polynomial of the Laplacian matrix of a graph 𝐺 of order 𝑛. In this paper, we give four transforms on graphs that decrease all Laplacian coefficients 𝑐_𝑘(𝐺) and investigate a conjecture A. Ilic and M. Ilic (2009) about the Laplacian coefficients of unicyclic graphs with 𝑛 vertices and 𝑚 pendent vertices. Finally, we determine the graph with the smallest Laplacian-like energy among all the unicyclic graphs with 𝑛 vertices and 𝑚 pendent vertices.

1. Introduction

Let 𝐺=(𝑉,𝐸) be a simple undirected graph with 𝑛 vertices and |𝐸| edges and, let 𝐿(𝐺)=𝐷(𝐺)−𝐴(𝐺) be its Laplacian matrix. The Laplacian polynomial of 𝐺 is the characteristic polynomial of its Laplacian matrix. That is Φ(𝐺,𝜆)=det𝜆𝐼_𝑛−𝐿(𝐺)=^𝑛_𝑘=0(−1)^𝑘𝑐_𝑘(𝐺)𝜆^𝑛−𝑘. (1.1)

The Laplacian matrix 𝐿(𝐺) has nonnegative eigenvalues 𝜇₁≥𝜇₂≥⋯≥𝜇_𝑛−1≥𝜇_𝑛=0 [1]. From Viette’s formulas, 𝑐_𝑘(𝐺)=𝜎_𝑘𝜇₁,𝜇₂,…,𝜇_𝑛−1=_{|𝐼|𝐼⊆{1,2,…,𝑛−1},=𝑘𝑖∈𝐼}𝜇_𝑖 (1.2) is a symmetric polynomial of order 𝑛−1. In particular, we have 𝑐₀(𝐺)=1,𝑐₁(𝐺)=2|𝐸(𝐺)|,𝑐_𝑛(𝐺)=0 and 𝑐_𝑛−1(𝐺)=𝑛𝜏(𝐺), where 𝜏(𝐺) is the number of spanning trees of 𝐺. If 𝐺 is a tree, coefficient 𝑐_𝑛−2(𝐺) is equal to its Wiener index, which is a sum of distance between all pairs of vertices: 𝑐_𝑛−2(𝐺)=𝑊(𝐺)=_{𝑢,𝑣∈𝑉}𝑑(𝑢,𝑣). (1.3) The Wiener index is considered as one of the most used topological indices with high correlation with many physical and chemical properties of molecular compounds.

A unicyclic graph is a connected graph in which the number of vertices equals the number of edges. Recently, the study on the Laplacian coefficients attracts much attention.

Mohar [2] proved that among all trees of order 𝑛, the 𝑘th Laplacian coefficients 𝑐_𝑘(𝐺) are largest when the tree is a path and are smallest for stars. Stevanovic and Ilic [3] showed that among all connected unicyclic graphs of order 𝑛, the 𝑘th Laplacian coefficients 𝑐_𝑘(𝐺) are largest when the graph is a cycle 𝐶_𝑛 and smallest when the graph is an 𝑆_𝑛 with an additional edge between two of its pendent vertices, where 𝑆_𝑛 is a star of order 𝑛. He and Shan [4] proved that among all bicyclic graphs of order 𝑛, the 𝑘th Laplacian coefficients 𝑐_𝑘(𝐺) is smallest when the graph is obtained from 𝐶₄ by adding one edge connecting two non-adjacent vertices and adding 𝑛−4 pendent vertices attached to the vertex of degree 3. A. Ilic and M. Ilic [5] verified that among trees on 𝑛 vertices and 𝑚 leaves, the balanced starlike tree 𝑆(𝑛,𝑚) (see Definition 2.2) has minimal Laplacian coefficients. Some other works on Laplacian coefficients can be found in [6–8].

In this paper, we determine the smallest 𝑘th Laplacian coefficients 𝑐_𝑘(𝐺) among all unicyclic graphs with 𝑛 vertices and 𝑚 pendent vertices. Thus we completely solve a conjecture on the minimal Laplacian coefficients of unicyclic graphs with 𝑛 vertices and 𝑚 pendent vertices (see [5]).

Motivated by the results in [3, 4, 9–12] concerning the minimal Laplacian coefficients and Laplacian-like energy of some graphs and the minimal molecular graph energy of unicyclic graphs with 𝑛 vertices and 𝑚 pendent vertices, this paper will characterize the unicyclic graphs with 𝑛 vertices and 𝑚 pendent vertices, which minimize Laplacian-like energy.

2. Transformations and Lemmas

In this section, we introduce some graphic transformations and lemmas, which can be used to prove our main results. The Laplacian coefficients 𝑐_𝑘(𝐺) of a graph 𝐺 can be expressed in terms of subtree structures of 𝐺 by the following result of Kelmans and Chelnokov [13]. Let 𝐹 be a spanning forest of 𝐺 with components 𝑇_𝑖, 𝑖=1,2,…,𝑘 having 𝑛_𝑖 vertices each, and let 𝛾(𝐹)=∏^𝑘_𝑖=1𝑛_𝑖.

Lemma 2.1 (see [13]). The Laplacian coefficient 𝑐_𝑛−𝑘(𝐺) of a graph 𝐺 is given by 𝑐_𝑛−𝑘(𝐺)=_{𝐹∈ℱ𝑘}𝛾(𝐹), (2.1) where ℱ_𝑘 is the set of all spanning forests of 𝐺 with exactly 𝑘 components.

For a real number 𝑥, we use ⌊𝑥⌋ to represent the largest integer not greater than 𝑥 and ⌈𝑥⌉ to represent the smallest integer not less than 𝑥.

Definition 2.2 (see [5]). The balanced starlike tree 𝑆(𝑛,𝑚), 3≤𝑚≤𝑛−1, is a tree of order 𝑛 with just one center vertex 𝑣, and each of the 𝑚 branches of 𝑇 at 𝑣 is a path of length ⌊(𝑛−1)/𝑚⌋ or ⌈(𝑛−1)/𝑚⌉.

Let 𝑃_𝑛 be the path with 𝑛 vertices. A path 𝑃∶𝑣𝑣₁𝑣₂⋯𝑣_𝑘 in 𝐺 is called a pendent path if 𝑑(𝑣₁)=𝑑(𝑣₂)=⋯=𝑑(𝑣_𝑘−1)=2 and 𝑑(𝑣_𝑘)=1. If 𝑘=1, then we say 𝑣𝑣₁ is a pendent edge of the graph 𝐺. A leaf or pendent vertex is a vertex of degree one. A branching vertex is a vertex of degree greater than two. The 𝑘 paths 𝑃_𝑙1,𝑃_𝑙2,…,𝑃_𝑙𝑘 are said to have almost equal lengths if 𝑙₁,𝑙₂,…,𝑙_𝑘 satisfy |𝑙_𝑖−𝑙_𝑗|≤1 for 1≤𝑖,𝑗≤𝑘.

Definition 2.3 (see [5]). The dumbbell 𝐷(𝑛,𝑎,𝑏) consists of the path 𝑃_{𝑛−𝑎−𝑏} together with 𝑎 independent vertices adjacent to one leaf of 𝑃_{𝑛−𝑎−𝑏} and 𝑏 independent vertices adjacent to the other leaf.

The union 𝐺=𝐺₁⋃𝐺₂ of graph 𝐺₁ and 𝐺₂ with disjoint vertex sets 𝑉₁ and 𝑉₂ and edge sets 𝐸₁ and 𝐸₂ is the graph 𝐺=(𝑉,𝐸) with 𝑉=𝑉₁⋃𝑉₂ and 𝐸=𝐸₁⋃𝐸₂. If 𝐺 is a union of two paths of lengths 𝑎 and 𝑏, then 𝐺 is disconnected and has 𝑎+𝑏 vertices and 𝑎+𝑏−2 edges. Let 𝑚_𝑘(𝐺) be the number of matchings of 𝐺 containing exactly 𝑘 independent edges. Especially, let 𝑚_𝑘(𝑎,𝑏) be the number of 𝑘 matchings in 𝐺=𝑃_𝑎⋃𝑃_𝑏.

Lemma 2.4 (see [5]). Let 𝑣 be a vertex of nontrivial connected graph 𝐺, and let 𝐺(𝑝,𝑞) denote the graph obtained from 𝐺 by adding pendent paths 𝑃=𝑣𝑣₁𝑣₂⋯𝑣_𝑝 and 𝑄=𝑣𝑢₁𝑢₂⋯𝑢_𝑞, at vertex 𝑣. Assume that both numbers 𝑝 and 𝑞 are even. If 𝑝−2≥𝑞+2≥4, then for every 𝑘 we have 𝑚_𝑘𝐺𝑝,𝑞≤𝑚_𝑘𝐺𝑝−2,𝑞+2. (2.2)

Lemma 2.5 (see [12]). Let 𝑚_𝑘(𝑎,𝑏) be the number of 𝑘-matchings in 𝐺=𝑃_𝑎⋃𝑃_𝑏 and 𝑛=4𝑠+𝑟 with 0≤𝑟≤3. Then the following inequality holds: 𝑚_𝑘(𝑛,0)≥𝑚_𝑘(𝑛−2,2)≥𝑚_𝑘(𝑛−4,4)≥⋯≥𝑚_𝑘(2𝑠+𝑟,2𝑠). (2.3)

Lemma 2.6 (see [5]). Among trees on 𝑛 vertices and 2≤𝑚≤𝑛−2 leaves, the balanced starlike tree 𝑆(𝑛,𝑚) has minimal Laplacian coefficient 𝑐_𝑘(𝐺), for every 𝑘=0,1,…,𝑛.

Definition 2.7 (see [5]). Let 𝑣 be a vertex of a tree 𝑇 of degree 𝑚+1. Suppose that 𝑃₁,𝑃₂,…,𝑃_𝑚 are pendent paths incident with 𝑣, with lengths 𝑛_𝑖≥1,𝑖=1,2,…,𝑚. Let 𝑤 be the neighbor of 𝑣 distinct from the starting vertices of paths 𝑣₁,𝑣₂,…,𝑣_𝑚, respectively. We form a tree 𝑇^′=𝛿(𝑇,𝑣) by removing the edges 𝑣𝑣₁,𝑣𝑣₂,…,𝑣𝑣_𝑚−1 from 𝑇 and adding 𝑚−1 new edges 𝑤𝑣₁,𝑤𝑣₂,…,𝑤𝑣_𝑚−1 incident with 𝑤. We say that 𝑇^′ is a 𝛿-transform of 𝑇.

Lemma 2.8 (see [5]). Let 𝑇 be an arbitrary tree, rooted at the center vertex. Let vertex 𝑣 be on the deepest level of tree 𝑇 among all branching vertices with degree at least three. Then for the 𝛿-transformation tree 𝑇^′=𝛿(𝑇,𝑣) and 0≤𝑘≤𝑛 holds: 𝑐_𝑘(𝑇)≥𝑐_𝑘𝑇^′. (2.4)

Lemma 2.9 (see [14]). For every acyclic graph 𝑇 with 𝑛 vertices, 𝑐_𝑘(𝑇)=𝑚_𝑘(𝑆(𝑇)),0≤𝑘≤𝑛, (2.5) where 𝑆(𝑇) means the subdivision graph of 𝑇.

3. Main Results

In this section, we present four new graphic transformations that decrease the Laplacian coefficients.

Definition 3.1. Let 𝑢 be a vertex in the cycle 𝐶 of a unicyclic graph 𝐺, such that 𝑢 has degree 𝑝+2 and 𝑝 pendent paths named 𝑃_𝑙1,𝑃_𝑙2,…,𝑃_𝑙𝑝, where 𝑃_𝑙𝑖:𝑢_𝑖,1,𝑢_𝑖,2,…,𝑢_{𝑖,𝑙𝑖},??1≤𝑖≤𝑝. If 𝑙_𝑖≥𝑙_𝑗+2, and let 𝐺₁=𝐺−𝑢_{𝑖,𝑙𝑖−1}𝑢_{𝑖,𝑙𝑖}+𝑢_{𝑗,𝑙𝑗}𝑢_{𝑖,𝑙𝑖}≜𝜋₁(𝐺). (3.1) We say that 𝐺₁ is a 𝜋₁-transformation of 𝐺.

It is easy to see that 𝜋₁-transformation preserves the size of a cycle of 𝐺 and the number of pendent vertices.

Theorem 3.2. Let 𝐺 be a connected unicyclic graph with 𝑛 vertices and 𝑚 pendent vertices, 𝐺₁=𝜋₁(𝐺). Then for every 𝑘=0,1,…,𝑛, 𝑐_𝑘(𝐺)≥𝑐_𝑘𝐺₁, (3.2) with equality if and only if 𝑘∈{0,1,𝑛−1,𝑛}.

Proof. It is easy to see that 𝑐₀(𝐺₁)=𝑐₀(𝐺)=1, 𝑐₁(𝐺₁)=2|𝐸(𝐺₁)|=2|𝐸(𝐺)|=𝑐₁(𝐺),𝑐_𝑛(𝐺₁)=𝑐_𝑛(𝐺)=0, 𝑐_𝑛−1(𝐺₁)=𝑛𝜏(𝐺₁)=𝑛|𝐸(𝐶)|=𝑛𝜏(𝐺)=𝑐_𝑛−1(𝐺).
Now, consider the coefficients 𝑐_𝑛−𝑘 (𝑘≠0,1,𝑛−1,𝑛). Let ℱ_𝑘 and ℱ_𝑘1 be the sets of spanning forests of 𝐺 and 𝐺₁ with exactly 𝑘 components, respectively.
Without loss of generality, we assume that 𝑙₁≥𝑙₂+2. Let 𝐺₁=𝜋₁(𝐺)=𝐺−𝑢_1,𝑙1−1𝑢_1,𝑙1+𝑢_2,𝑙2𝑢_1,𝑙1 (see Figure 1).
Obviously, by the definition of the spanning forest, the cycle 𝐶 in the unicyclic graph satisfies that 𝐶∉𝐹∈ℱ_𝑘 and 𝐶∉𝐹₁∈ℱ_𝑘1, where 𝐹 and 𝐹₁ are the arbitrary forests in ℱ_𝑘 and ℱ_𝑘1, respectively. Without loss of generality, we remove one of the edges in the cycle 𝐶, say 𝑢𝑣, so we get 𝑇 and 𝑇^′, respectively. By Lemmas 2.4 and 2.9, we have that for every 𝑘=0,1,…,𝑛, 𝑐_𝑘(𝑇)≥𝑐_𝑘𝑇^′, (3.3) with equality if and only if 𝑘∈{0,1,𝑛−1,𝑛}. If we remove the other edge, say 𝑥𝑦, we get 𝑆 and 𝑆^′, respectively. By Lemmas 2.4 and 2.9, we have that for every 𝑘=0,1,…,𝑛, 𝑐_𝑘(𝑆)≥𝑐_𝑘𝑆^′, (3.4) with equality if and only if 𝑘∈{0,1,𝑛−1,𝑛}.
It is easy to see that 𝑇−𝑥𝑦=𝑆−𝑢𝑣 and 𝑇^′−𝑥𝑦=𝑆^′−𝑢𝑣. We know that the numbers of the same tree of spanning forests of 𝑇−𝑥𝑦 and 𝑇^′−𝑥𝑦 with exactly 𝑘 components are equal to the numbers of the same tree of spanning forests of 𝑆−𝑢𝑣 and 𝑆^′−𝑢𝑣 with exactly 𝑘 components, respectively.
Applying to Definition 3.1 and Lemma 2.1, we can show that for every 𝑘=0,1,…,𝑛, 𝑐_𝑘(𝐺)≥𝑐_𝑘𝐺₁, (3.5) with equality if and only if 𝑘∈{0,1,𝑛−1,𝑛}.

Figure 1 

Example of 𝜋1-transformation.

Definition 3.3. Let 𝑣 be a vertex in a cycle 𝐶 of a connected unicyclic graph 𝐺, where 𝑑(𝑣)≥3. Suppose that 𝑢 is one of two neighbors adjacent to 𝑣 in 𝐶, such that 𝑢 has degree 𝑝+2 and 𝑝 pendent paths incident with 𝑢 and 𝑣 has degree 𝑞+2 and 𝑞 pendent paths incident with 𝑣. Let 𝐺₂=𝐺−𝑣𝑣_𝑞+1+𝑢𝑣_𝑞+1≜𝜋₂(𝐺), (3.6) where 𝑣_𝑞+1 is one of the other neighbors adjacent to 𝑣 in 𝐶. We say that 𝐺₂ is a 𝜋₂-transformation of 𝐺 (see Figure 2).

Figure 2 

Example of 𝜋2-transformation.

Obviously, 𝜋₂-transformation decreases the size of a cycle of 𝐺 and preserves the number of pendent vertices.

Theorem 3.4. Let 𝐺 be a connected unicyclic graph with 𝑛 vertices and 𝑚 pendent vertices, 𝐺₂=𝜋₂(𝐺). Then for every 𝑘=0,1,…,𝑛, 𝑐_𝑘(𝐺)≥𝑐_𝑘𝐺₂, (3.7) with equality if and only if 𝑘∈{0,1,𝑛}.

Proof. Obviously, 𝑐₀(𝐺₂)=𝑐₀(𝐺)=1,𝑐₁(𝐺₂)=2|𝐸(𝐺₂)|=2|𝐸(𝐺)|=𝑐₁(𝐺),𝑐_𝑛(𝐺₂)=𝑐_𝑛(𝐺)=0. For 𝑘=𝑛−1, the length of a cycle in 𝐺 is greater than the length of a cycle in 𝐺₂. Therefore, 𝑐_𝑛−1(𝐺)>𝑐_𝑛−1(𝐺₂).
Now, consider the coefficients 𝑐_𝑛−𝑘 (𝑘≠0,1,𝑛−1,𝑛). Let ℱ_𝑘 and ℱ_𝑘2 be the sets of spanning forests of 𝐺 and 𝐺₂ with exactly 𝑘 components, respectively. Let 𝐹₂∈ℱ_𝑘2 and 𝑇^′ be the component of 𝐹₂ and 𝑢∈𝑉(𝑇^′). If 𝑣_𝑞+1∈𝑉(𝑇^′), we define 𝐹 with 𝑉(𝐹)=𝑉(𝐺) and 𝐸(𝐹)=𝐸𝐹₂−𝑢𝑣_𝑞+1+𝑣𝑣_𝑞+1. (3.8) Now, we distinguish 𝐹₂ as the following two cases.
Case??1 (𝑣∈𝑉(𝑇^′)). We have trees of equal sizes in both spanning forests thus 𝛾(𝐹)=𝛾(𝐹₂).
Case??2 (𝑣∉𝑉(𝑇^′)). Let vertex 𝑣 be in the tree 𝑆^′, that is, 𝑣∈𝑉(𝑆^′).
Note the fact that 𝑢𝑣 is a cut edge of 𝐺₂. It is easy to see that 𝐹 is a spanning forest of 𝐺, and the number of components of 𝐹 is 𝑘−1 or 𝑘. We claim that 𝐹∈ℱ_𝑘. Otherwise, 𝑢,𝑣 belong to one tree of 𝐹; then there exists a path 𝑃 joining 𝑣_𝑞+1 to 𝑢 in 𝐹; then 𝑢𝑃𝑣_𝑞+1𝑢 is a cycle of 𝐹₂, which contradicts the fact that 𝐹₂ is a forest.
Suppose that 𝑇^′−𝑣_𝑞+1 contains 𝑎≥1 vertices in the cycle 𝐶 (including 𝑢) and 𝑏≥0 vertices in the paths 𝑃₁,…,𝑃_𝑝, and 𝑇^′−𝑢 contains 𝑐≥1 vertices in the cycle 𝐶. Let 𝑆^′ contain 𝑑≥1 in the paths 𝑃_𝑝+1,…,𝑃_𝑝+𝑞. Assume the orders of the components of 𝐹₂ different from 𝑇^′ and 𝑆^′ are 𝑛₁,𝑛₂,…𝑛_𝑘−2. We have 𝛾(𝐹)−𝛾𝐹₂=[(𝑎+𝑏)(𝑐+𝑑)−(𝑎+𝑏+𝑐)𝑑]^𝑘−2_𝑖=1𝑛_𝑖=𝑐(𝑎+𝑏−𝑑)^𝑘−2_𝑖=1𝑛_𝑖=𝑐(𝑎+𝑏−𝑑)𝑁, (3.9) where 𝑁=∏^𝑘−2_𝑖=1𝑛_𝑖.
If we sum all differences for such forest, having fixed values 𝑎,𝑐 and 𝑏+𝑑=𝑀, we get _{𝐹∈ℱ∗}𝛾(𝐹)−𝛾𝐹₂=_{𝐹∈ℱ∗}𝑐(𝑎+𝑏−𝑑)𝑁=𝑐𝑁^𝑀−1_𝑏=0(𝑎+2𝑏−𝑀)=(𝑎−1)𝑐𝑁𝑀. (3.10)
It is easy to see that 𝑎≥1 and 𝑐≥1, so (𝑎−1)𝑐𝑁𝑀≥0. Since at least one vertex is in 𝐶−𝑢−𝑣_𝑞+1, there exists one forest 𝐹₂ such that 𝑎>1 and 𝑐≥1, and then (𝑎−1)𝑐𝑁𝑀>0.
If 𝑣_𝑞+1∉𝑉(𝑇^′), thus 𝛾(𝐹)=𝛾(𝐹₂).
Therefore, by using Lemma 2.1, we get 𝑐_𝑘(𝐺)=_{𝐹∈ℱ𝑘}𝛾(𝐹)>_{𝐹2∈ℱ𝑘2}𝛾𝐹₂=𝑐_𝑘𝐺₂. (3.11) This completes the proof of Theorem 3.4.

Definition 3.5. Let 𝑣 (not in the cycle 𝐶) be a vertex of degree 𝑞+1 in a connected unicyclic graph G. Suppose that 𝑃_𝑝+1,…,𝑃_𝑝+𝑞 are pendent paths incident with 𝑣. Let 𝑢 be the neighbor of 𝑣 distinct from the starting vertices of paths 𝑣₁,𝑣₂,…,𝑣_𝑞, respectively. Let 𝐺₃=𝜋₃(𝐺)=𝐺−𝑣𝑣₂−𝑣𝑣₃−⋯−𝑣𝑣_𝑞+𝑢𝑣₂+𝑢𝑣₃+⋯+𝑢𝑣_𝑞. (3.12) We say that 𝐺₃ is a 𝜋₃-transformation of 𝐺 (see Figure 3).

Figure 3 

Example of 𝜋3-transformation.

It is not difficult to see that 𝜋₃-transformation preserves the size of a cycle of 𝐺 and the number of pendent vertices.

Theorem 3.6. Let 𝐺 be a connected unicyclic graph with 𝑛 vertices and 𝑚 pendent vertices, 𝐺₃=𝜋₃(𝐺). Then for every 𝑘=0,1,…,𝑛, 𝑐_𝑘(𝐺)≥𝑐_𝑘𝐺₃, (3.13) with equality if and only if 𝑘∈{0,1,𝑛−1,𝑛}.

Proof. Obviously, 𝑐₀(𝐺₃)=𝑐₀(𝐺)=1,𝑐₁(𝐺₃)=2|𝐸(𝐺₃)|=2|𝐸(𝐺)|=𝑐₁(𝐺),𝑐_𝑛(𝐺₃)=𝑐_𝑛(𝐺)=0, 𝑐_𝑛−1(𝐺₃)=𝑛𝜏(𝐺₃)=𝑛|𝐸(𝐶)|=𝑛𝜏(𝐺)=𝑐_𝑛−1(𝐺).
Now, consider the coefficients 𝑐_𝑛−𝑘 (𝑘≠0,1,𝑛−1,𝑛). Let ℱ_𝑘 and ℱ_𝑘3 be the sets of spanning forests of 𝐺 and 𝐺₃ with exactly 𝑘 components, respectively. Obviously, by the definition of the spanning forest, the cycle 𝐶 in the unicyclic graph satisfies that 𝐶∉𝐹∈ℱ_𝑘 and 𝐶∉𝐹₃∈ℱ_𝑘3, where 𝐹 and 𝐹₃ are the arbitrary forests in ℱ_𝑘 and ℱ_𝑘3, respectively. Without loss of generality, we remove one of the edges on the cycle, say 𝑤𝑢, so we get two trees 𝑇 and 𝑇^′, respectively. Applying to Definition 2.7, we know that 𝑇^′=𝛿(𝑇). Then using Lemma 2.8, we can get that for every 𝑘=0,1,…,𝑛, 𝑐_𝑘(𝑇)≥𝑐_𝑘𝑇^′, (3.14) with equality if and only if 𝑘∈{0,1,𝑛−1,𝑛}. If we remove another edge, say 𝑥𝑦, we get 𝑆 and 𝑆^′, respectively. By Definition 2.7, we know that 𝑆^′=𝛿(𝑆). Then applying to Lemma 2.8, we get that for every 𝑘=0,1,…,𝑛, 𝑐_𝑘(𝑆)≥𝑐_𝑘𝑆^′, (3.15) with equality if and only if 𝑘∈{0,1,𝑛−1,𝑛}.
It is easy to see that 𝑇−𝑥𝑦=𝑆−𝑢𝑣 and 𝑇^′−𝑥𝑦=𝑆^′−𝑢𝑣. We know that the numbers of the same tree of spanning forests of 𝑇−𝑥𝑦 and 𝑇^′−𝑥𝑦 with exactly 𝑘 components are equal to the numbers of the same tree of spanning forests of 𝑆−𝑢𝑣 and 𝑆^′−𝑢𝑣 with exactly 𝑘 components, respectively.
By Definition 3.5 and Lemma 2.1, we have that for every 𝑘=0,1,…,𝑛, 𝑐_𝑘(𝐺)≥𝑐_𝑘𝐺₃, (3.16) with equality if and only if 𝑘∈{0,1,𝑛−1,𝑛}.

Definition 3.7. Let 𝑢, 𝑣, and 𝑤 be three vertices on the triangle in a unicyclic graph 𝐺. Suppose that 𝑃₁,…,𝑃_𝑝 are pendent paths incident with 𝑢, 𝑃_𝑝+1,…,𝑃_𝑝+𝑞 are pendent paths incident with 𝑣, and 𝑃_𝑝+𝑞+1,…,𝑃_{𝑝+𝑞+𝑙} are pendent paths incident with 𝑤(𝑝+𝑞+𝑙=𝑚). Let 𝐺₄=𝐺−𝑣𝑣₁−⋯−𝑣𝑣_𝑞+𝑢𝑣₁+⋯+𝑢𝑣_𝑞≜𝜋₄(𝐺). (3.17) We say that 𝐺₄ is a 𝜋₄-transformation of 𝐺 (see Figure 4).

Figure 4 

Example of 𝜋4-transformation.

Theorem 3.8. Let 𝑢, 𝑣, and 𝑤 be three vertices on the triangle in a unicyclic graph 𝐺, 𝐺₄=𝜋₄(𝐺). Then for every 𝑘=0,1,…,𝑛, 𝑐_𝑘(𝐺)≥𝑐_𝑘𝐺₄, (3.18) with equality if and only if 𝑘∈{0,1,𝑛−1,𝑛}.

Proof. It is obvious to see that 𝑐₀(𝐺₄)=𝑐₀(𝐺)=1,𝑐₁(𝐺₄)=2|𝐸(𝐺₄)|=2|𝐸(𝐺)|=𝑐₁(𝐺),𝑐_𝑛(𝐺₄)=𝑐_𝑛(𝐺)=0. For 𝑘=𝑛−1, the length of a cycle in 𝐺₄ is equal to the length of a cycle in 𝐺. Therefore, 𝑐_𝑛−1(𝐺)=𝑐_𝑛−1(𝐺₄).
Now, consider the coefficient 𝑐_𝑛−𝑘 (𝑘≠0,1,𝑛−1,𝑛). Let ℱ_𝑘 and ℱ_𝑘4 be the sets of spanning forests of 𝐺 and 𝐺₄ with exactly 𝑘 components, respectively.
Similarly to the proof of Theorem 3.2, we can get 6 trees as shown in Figure 5. Obviously, by Definition 2.7, we know that 𝑇^′_𝑖=𝛿(𝑇_𝑖) (𝑖=1,2,3). And according to Lemma 2.8, we can verify that 𝑐_𝑘𝑇₁≥𝑐_𝑘𝑇^′₁,𝑐_𝑘𝑇₂≥𝑐_𝑘𝑇^′₂,𝑐_𝑘𝑇₃≥𝑐_𝑘𝑇^′₃. (3.19) By , Definition 3.7, and Lemma 2.1, it is easy to see that for every 𝑘=0,1,…,𝑛, 𝑐_𝑘(𝐺)≥𝑐_𝑘𝐺₄, (3.20) with equality if and only if 𝑘∈{0,1,𝑛−1,𝑛}. This completes the proof of Theorem 3.8.

Figure 5 

Obtained trees from Figure 4.

Theorem 3.9. Let 𝐺 be a connected unicyclic graph with 𝑛 vertices and 𝑚 pendent vertices. Then for 0≤𝑘≤𝑛, 𝑐_𝑘(𝐺)≥𝑐_𝑘𝑆^′(𝑛,𝑚), (3.21) with equality if and only if 𝑘∈{0,1,𝑛}, where 𝑆^′(𝑛,𝑚) is as shown in Figure 6, and each of the 𝑚 branches at 𝑣 is a path of length ⌊(𝑛−3)/𝑚⌋ or ⌈(𝑛−3)/𝑚⌉.

Figure 6 

𝑆′(𝑛,𝑚).

Proof. Let 𝐶=𝑤₁𝑤₂⋯𝑤_𝑡𝑤₁ be a cycle of connected unicyclic graph 𝐺, and let 𝑇_𝑖 be a tree attached at 𝑤_𝑖, 𝑖=1,2,…,𝑡. We can apply 𝜋₃-transformation to 𝑇_𝑖, such that the tree contains one branch vertex 𝑤_𝑖 with pendent path attached to it. Next, we can apply 𝜋₂-transformation to decrease the size of the cycle 𝐶 as long as the length of 𝐶 is not 3. Then we can apply 𝜋₁-transformation at the longest and the shortest path repeatedly, the Laplacian coefficients do not increase while the attached paths become more balanced. Finally, we can apply 𝜋₄-transformation as long as it is not 𝑆^′(𝑛,𝑚).
According to Theorems 3.2, 3.4, 3.6, and 3.8, we know that 𝜋_𝑖-transformation (𝑖=1,2,3,4) cannot increase the Laplacian coefficients. So, for an arbitrary unicyclic graph 𝐺 with 𝑛 vertices and 𝑚 pendent vertices, we verify that 𝑐_𝑘(𝐺)≥𝑐_𝑘𝑆^′(𝑛,𝑚), (3.22) where 0≤𝑘≤𝑛 and with equality if and only if 𝑘=0,1,𝑛. This completes the proof of Theorem 3.9.

4. Laplacian-Like Energy of Unicyclic Graphs with 𝑚 Pendent Vertices

Let 𝐺 be a graph. The Laplacian-like energy of graph 𝐺, LEL for short, is defined as follows: LEL(𝐺)=^𝑛−1_𝑘=1√𝜇_𝑘, (4.1) where 𝜇₁≥𝜇₂≥⋯≥𝜇_𝑛=0 are the Laplacian eigenvalues of 𝐺. This concept was introduced by J. Liu and B. Liu [9], where it was demonstrated it has similar feature as molecular graph energy (for more details see [15]). Stevanovic in [10] presented a connection between LEL and Laplacian coefficients.

Theorem 4.1. Let 𝐺 and 𝐻 be two graphs with 𝑛 vertices. If 𝑐_𝑘(𝐺)≤𝑐_𝑘(𝐻) for 𝑘=1,2,…,𝑛−1, then LEL(𝐺)≤LEL(𝐻). Furthermore, if a strict inequality 𝑐_𝑘(𝐺)<𝑐_𝑘(𝐻) holds for some 1≤𝑘≤𝑛−1, then LEL(𝐺)<LEL(𝐻).

Using this result, we can conclude the following.

Corollary 4.2. Let 𝐺 be a connected unicyclic graph with 𝑛 vertices and 𝑚 pendent vertices. Then if 𝐺≇𝑆^′(𝑛,𝑚)LEL𝑆^′(𝑛,𝑚)<LEL(𝐺), (4.2) where 𝑆^′(𝑛,𝑚) is shown in Figure 6, and each of the 𝑚 branches at 𝑣 is a path of length ⌊(𝑛−3)/𝑚⌋ or ⌈(𝑛−3)/𝑚⌉.

Acknowledgment

The authors would like to express their sincere gratitude to the anonymous referees whose constructive comments, valuable suggestions, and careful reading improved the final form of this paper.