On the Signless Laplacian Spectral Radius of Bicyclic Graphs with Perfect Matchings

Abstract

The graph with the largest signless Laplacian spectral radius among all bicyclic graphs with perfect matchings is determined.

1. Introduction

Let 𝐺=(𝑉,𝐸) be a simple connected graph with vertex set 𝑉={𝑣₁,𝑣₂,…,𝑣_𝑛} and edge set 𝐸. Its adjacency matrix 𝐴(𝐺)=(𝑎_𝑖𝑗) is defined as 𝑛×𝑛 matrix (𝑎_𝑖𝑗), where 𝑎_𝑖𝑗=1 if 𝑣_𝑖 is adjacent to 𝑣_𝑗, and 𝑎_𝑖𝑗=0, otherwise. Denote by 𝑑(𝑣_𝑖) or 𝑑_𝐺(𝑣_𝑖) the degree of the vertex 𝑣_𝑖. Let 𝑄(𝐺)=𝐷(𝐺)+𝐴(𝐺) be the signless Laplacian matrix of graph 𝐺, where 𝐷(𝐺)=diag(𝑑(𝑣₁),𝑑(𝑣₂),…,𝑑(𝑣_𝑛)) denotes the diagonal matrix of vertex degrees of 𝐺. It is well known that 𝐴(𝐺) is a real symmetric matrix and 𝑄(𝐺) is a positive semidefinite matrix. The largest eigenvalues of 𝐴(𝐺) and 𝑄(𝐺) are called the spectral radius and the signless Laplacian spectral radius of 𝐺, denoted by 𝜌(𝐺) and 𝑞(𝐺), respectively. When 𝐺 is connected, 𝐴(𝐺) and 𝑄(𝐺) are a nonnegative irreducible matrix. By the well-known Perron-Frobenius theory, 𝜌(𝐺) is simple and has a unique positive unit eigenvector and so does 𝑞(𝐺). We refer to such an eigenvector corresponding to 𝑞(𝐺) as the Perron vector of 𝐺.

Two distinct edges in a graph 𝐺 are independent if they are not adjacent in 𝐺. A set of mutually independent edges of 𝐺 is called a matching of 𝐺. A matching of maximum cardinality is a maximum matching in 𝐺. A matching 𝑀 that satisfies 2|𝑀|=𝑛=|𝑉(𝐺)| is called a perfect matching of the graph 𝐺. Denote by 𝐶_𝑛 and 𝑃_𝑛 the cycle and the path on 𝑛 vertices, respectively.

The characteristic polynomial of 𝐴(𝐺) is det(𝑥𝐼−𝐴(𝐺)), which is denoted by Φ(𝐺) or Φ(𝐺,𝑥). The characteristic polynomial of 𝑄(𝐺) is det(𝑥𝐼−𝑄(𝐺)), which is denoted by Ψ(𝐺) or Ψ(𝐺,𝑥).

A bicyclic graph is a connected graph in which the number of vertices equals the number of edges minus one. Let 𝐶_𝑝 and 𝐶_𝑞 be two vertex-disjoint cycles. Suppose that 𝑣₁ is a vertex of 𝐶_𝑝 and 𝑣_𝑙 is a vertex of 𝐶_𝑞. Joining 𝑣₁ and 𝑣_𝑙 by a path 𝑣₁𝑣₂⋯𝑣_𝑙 of length 𝑙−1, where 𝑙≥1 and 𝑙=1 means identifying 𝑣₁ with 𝑣_𝑙, denoted by 𝐵(𝑝,𝑙,𝑞), is called an ∞-graph (see Figure 1). Let 𝑃_𝑙+1,𝑃_𝑝+1, and 𝑃_𝑞+1 be the three vertex-disjoint paths, where 𝑙,𝑝,𝑞≥1, and at most one of them is 1. Identifying the three initial vertices and the three terminal vertices of them, respectively, denoted by 𝑃(𝑙,𝑝,𝑞), is called a 𝜃-graph (see Figure 2).

Figure 1 

𝐵(𝑝,1,𝑞) and 𝐵(𝑝,𝑙,𝑞)(𝑙≥2).

Figure 2 

𝑃(𝑝,𝑙,𝑞).

Let 𝐵_𝑛(2𝜇) be the set of all bicyclic graphs on 𝑛=2𝜇(𝜇≥2) vertices with perfect matchings. Obviously 𝐵_𝑛(2𝜇) consists of two types of graphs: one type, denoted by 𝐵⁺_𝑛(2𝜇), is a set of graphs each of which is an ∞-graph with trees attached; the other type, denoted by 𝐵_𝑛⁺⁺(2𝜇), is a set of graphs each of which is 𝜃- graph with trees attached. Then we have 𝐵_𝑛(2𝜇)=𝐵⁺_𝑛(2𝜇)∪𝐵_𝑛⁺⁺(2𝜇).

The investigation on the spectral radius of graphs is an important topic in the theory of graph spectra, in which some early results can go back to the very beginnings (see [1]). The recent developments on this topic also involve the problem concerning graphs with maximal or minimal spectral radius of a given class of graphs. In [2], Chang and Tian gave the first two spectral radii of unicyclic graphs with perfect matchings. Recently, Yu and Tian [3] gave the first two spectral radii of unicyclic graphs with a given matching number; Guo [4] gave the first six spectral radii over the class of unicyclic graphs on a given number of vertices; and Guo [5] gave the first ten spectral radii over the class of unicyclic graphs on a given number of vertices and the first four spectral radii of unicyclic graphs with perfect matchings. For more results on this topic, the reader is referred to [6–9] and the references therein.

In this paper, we deal with the extremal signless Laplacian spectral radius problems for the bicyclic graphs with perfect matchings. The graph with the largest signless Laplacian spectral radius among all bicyclic graphs with perfect matchings is determined.

2. Lemmas

Let 𝐺−𝑢 or 𝐺−𝑢𝑣 denote the graph obtained from 𝐺 by deleting the vertex 𝑢∈𝑉(𝐺) or the edge 𝑢𝑣∈𝐸(𝐺). A pendant vertex of 𝐺 is a vertex with degree 1. A path 𝑃∶𝑣𝑣₁𝑣₂⋯𝑣_𝑘 in 𝐺 is called a pendant 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 pendant edge of the graph 𝐺.

In order to complete the proof of our main result, we need the following lemmas.

Lemma 1 (see [10, 11]). Let 𝐺 be a connected graph and 𝑢,𝑣 two vertices of 𝐺. Suppose that 𝑣₁,𝑣₂,…,𝑣_𝑠∈𝑁(𝑣)⧵{𝑁(𝑢)∪𝑢}(1≤𝑠≤𝑑(𝑣)) and 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛) is the Perron vector of 𝐺, where 𝑥_𝑖 corresponds to the vertex 𝑣_𝑖(1≤𝑖≤𝑛). Let 𝐺^∗ be the graph obtained from 𝐺 by deleting the edges 𝑣𝑣_𝑖 and adding the edges 𝑢𝑣_𝑖(1≤𝑖≤𝑠). If 𝑥_𝑢≥𝑥_𝑣, then 𝑞(𝐺)<𝑞(𝐺^∗).

The cardinality of a maximum matching of 𝐺 is commonly known as its matching number, denoted by 𝜇(𝐺).

From Lemma 1, we have the following results.

Corollary 2. Let 𝑤 and 𝑣 be two vertices in a connected graph 𝐺 and suppose that 𝑠 paths {𝑤𝑤₁𝑤^′₁,𝑤𝑤₂𝑤^′₂,…,𝑤𝑤_𝑠𝑤^′_𝑠} of length 2 are attached to 𝐺 at 𝑤 and 𝑡 paths {𝑣𝑣₁𝑣^′₁,𝑣𝑣₂𝑣^′₂,…,𝑣𝑣_𝑡𝑣^′_𝑡} of length 2 are attached to 𝐺 at 𝑣 to form 𝐺_𝑠,𝑡. Then either 𝑞(𝐺_{𝑠+𝑖,𝑡−𝑖})>𝑞(𝐺_𝑠,𝑡)(1≤𝑖≤𝑡) or 𝑞(𝐺_{𝑠−𝑖,𝑡+𝑖})>𝑞(𝐺_𝑠,𝑡)(1≤𝑖≤𝑠)or 𝜇(𝐺_0,𝑠+𝑡)=𝜇(𝐺_𝑠+𝑡,0)=𝜇(𝐺_𝑠,𝑡).

Corollary 3. Suppose 𝑢 is a vertex of graph 𝐺 with 𝑑(𝑢)≥2. Let 𝐺∶𝑢𝑣 be a graph obtained by attaching a pendant edge 𝑢𝑣 to 𝐺 at 𝑢. Suppose 𝑡 paths {𝑣𝑣₁𝑣^′₁,…,𝑣𝑣_𝑡𝑣^′_𝑡} of length 2 are attached to 𝐺∶𝑢𝑣 at 𝑣 to form 𝐿_0,𝑡. Let 𝑀_1,𝑡=𝐿_0,𝑡−𝑣𝑣₁−⋯−𝑣𝑣_𝑡+𝑢𝑣₁+⋯+𝑢𝑣_𝑡. (1) If 𝐿_0,𝑡 has a perfect matching, then we have that 𝑀_1,𝑡 has a perfect matching and 𝑞𝑀_1,𝑡>𝑞𝐿_0,𝑡,(𝑡≥1). (2)

An internal path of a graph 𝐺 is a sequence of vertices 𝑣₁,𝑣₂,…,𝑣_𝑚 with 𝑚≥2 such that(1)the vertices in the sequences are distinct (except possibly 𝑣₁=𝑣_𝑚);(2)𝑣_𝑖 is adjacent to 𝑣_𝑖+1,(𝑖=1,2,…,𝑚−1);(3)the vertex degrees 𝑑(𝑣_𝑖) satisfy 𝑑(𝑣₁)≥3, 𝑑(𝑣₂)=⋯=𝑑(𝑣_𝑚−1)=2 (unless 𝑚=2) and 𝑑(𝑣_𝑚)≥3.

Let 𝐺 be a connected graph, and 𝑢𝑣∈𝐸(𝐺). The graph 𝐺_𝑢𝑣 is obtained from 𝐺 by subdividing the edge 𝑢𝑣, that is, adding a new vertex 𝑤 and edges 𝑢𝑤,𝑤𝑣 in 𝐺−𝑢𝑣. By similar reasoning as that of Theorem 3.1 of [12], we have the following result.

Lemma 4. Let 𝑃∶𝑣₁𝑣₂⋯𝑣_𝑘(𝑘≥2) be an internal path of a connected graph 𝐺. Let 𝐺^′ be a graph obtained from 𝐺 by subdividing some edge of 𝑃. Then we have 𝑞(𝐺^′)<𝑞(𝐺).

Corollary 5. Suppose that 𝑣₁𝑣₂⋯𝑣_𝑘(𝑘≥3) is an internal path of the graph 𝐺 and 𝑣₁𝑣_𝑘∉𝐸(𝐺) for 𝑘=3. Let 𝐺^∗ be the graph obtained from 𝐺−𝑣_𝑖𝑣_𝑖+1−𝑣_𝑖+1𝑣_𝑖+2(1≤𝑖≤𝑘−2) by amalgamating 𝑣_𝑖, 𝑣_𝑖+1, and 𝑣_𝑖+2 to form a new vertex 𝑤₁ together with attaching a new pendant path 𝑤₁𝑤₂𝑤₃ of length 2 at 𝑤₁. Then 𝑞(𝐺^∗)>𝑞(𝐺) and 𝜇(𝐺^∗)≥𝜇(𝐺).

Proof. From Lemma 4 and the well-known Perron-Frobenius theorem, It is easy to prove that 𝑞(𝐺^∗)>𝑞(𝐺). Next, we prove that 𝜇(𝐺^∗)≥𝜇(𝐺). Let 𝑀 be a maximum matching of 𝐺. If 𝑣_𝑖𝑣_𝑖+1∈𝑀 or 𝑣_𝑖+1𝑣_𝑖+2∈𝑀, then {𝑀−{𝑣_𝑖𝑣_𝑖+1}}∪{𝑤₂𝑤₃} or {𝑀−{𝑣_𝑖+1𝑣_𝑖+2}}∪{𝑤₂𝑤₃} is a matching of 𝐺^∗. Thus, 𝜇(𝐺^∗)≥𝜇(𝐺); If 𝑣_𝑖𝑣_𝑖+1∉𝑀 and 𝑣_𝑖+1𝑣_𝑖+2∉𝑀, then there exist two edges 𝑣_𝑖𝑢 and 𝑣_𝑖+2𝑣∈𝑀. Thus, {𝑀−{𝑣_𝑖𝑢}}∪{𝑤₂𝑤₃} is a matching of 𝐺^∗. Hence, 𝜇(𝐺^∗)≥𝜇(𝐺), completing the proof.

Let 𝑆(𝐺) be the subdivision graph of 𝐺 obtained by subdividing every edge of 𝐺.

Lemma 6 (see [13, 14]). Let 𝐺 be a graph on 𝑛 vertices and 𝑚 edges, Φ(𝐺)=det(𝑥𝐼−𝐴(𝐺)), Ψ(𝐺)=det(𝑥𝐼−𝑄(𝐺)). Then Φ(𝑆(𝐺))=𝑥^𝑚−𝑛Ψ(𝐺,𝑥²).

Lemma 7 (see [15]). Let 𝑢 be a vertex of a connected graph 𝐺. Let 𝐺_𝑘,𝑙(𝑘,𝑙≥0) be the graph obtained from 𝐺 by attaching two pendant paths of lengths 𝑘 and 𝑙 at 𝑢, respectively. If 𝑘≥𝑙≥1, then 𝑞(𝐺_𝑘,𝑙)>𝑞(𝐺_{𝑘+1,𝑙−1}).

Corollary 8. Suppose that 𝑣₁𝑣₂⋯𝑣_𝑘(𝑘≥3) is a pendant path of the graph 𝐺 with 𝑑(𝑣₁)≥3. Let 𝐺^∗ be the graph obtained from 𝐺−𝑣₁𝑣₂−𝑣₂𝑣₃ by amalgamating 𝑣₁, 𝑣₂, and 𝑣₃ to form a new vertex 𝑤₁ together with attaching a new pendant path 𝑤₁𝑤₂𝑤₃ of length 2 at 𝑤₁. Then 𝑞(𝐺^∗)>𝑞(𝐺) and 𝜇(𝐺^∗)≥𝜇(𝐺).

Proof. By Lemma 7 we have 𝑞(𝐺^∗)>𝑞(𝐺). By the proof as that of Corollary 5, we have 𝜇(𝐺^∗)≥𝜇(𝐺).

Lemma 9 (see [16]). Let 𝑒=𝑢𝑣 be an edge of 𝐺, and let 𝐶(𝑒) be the set of all circuits containing 𝑒. Then Φ(𝐺) satisfies Φ(𝐺)=Φ(𝐺−𝑒)−Φ(𝐺−𝑢−𝑣)−2_𝑍Φ(𝐺−𝑉(𝑍)), (3) where the summation extends over all 𝑍∈𝐶(𝑒).

Lemma 10 (see [16]). Let 𝑣 be a vertex of 𝐺, and let 𝜑(𝑣) be the collection of circuits containing 𝑣, and let 𝑉(𝑍) denote the set of vertices in the circuit 𝑍. Then the characteristic polynomial Φ(𝐺) satisfies Φ(𝐺)=𝑥Φ(𝐺−𝑣)−_𝑤Φ(𝐺−𝑣−𝑤)−2_{(𝑣)𝑍∈𝜑}Φ(𝐺−𝑉(𝑍)), (4) where the first summation extends over those vertices 𝑤 adjacent to 𝑣, and the second summation extends over all 𝑍∈𝜑(𝑣).

Lemma 11 (see [17]). Let 𝐺 be a connected graph, and let 𝐺^′ be a proper spanning subgraph of 𝐺. Then 𝜌(𝐺)>𝜌(𝐺^′), and, for 𝑥≥𝜌(𝐺),Φ(𝐺^′)>Φ(𝐺).

Let Δ(𝐺) denote the maximum degree of 𝐺. From Lemma 11, we have 𝜌(𝐺)≥Δ(𝐺).

Lemma 12 (see [13]). Let 𝐺 be a connected graph, and let 𝐺^′ be a proper spanning subgraph of 𝐺. Then 𝑞(𝐺)>𝑞(𝐺^′).

Lemma 13 (see [18]). Let 𝐺=(𝑉,𝐸) be a connected graph with vertex set 𝑉={𝑣₁,𝑣₂,…,𝑣_𝑛}. Suppose that 𝑣₁𝑣₂∈𝐸(𝐺), 𝑣₁𝑣₃∈𝐸(𝐺), 𝑣₁𝑣₄∈𝐸(𝐺), 𝑑(𝑣₃)≥2, 𝑑(𝑣₄)≥2, 𝑑(𝑣₁)=3, and 𝑑(𝑣₂)=1. Let 𝐺_𝑣1𝑣3(𝐺_𝑣1𝑣4) be the graph obtained from 𝐺−𝑣₁𝑣₃(𝐺−𝑣₁𝑣₄) by amalgamating 𝑣₁ and 𝑣₃(𝑣₄) to form a new vertex 𝑤₁(𝑤₃) together with subdivising the edge 𝑤₁𝑣₂(𝑤₃𝑣₂) with a new vertex 𝑤₂(𝑤₄). If 𝑞=𝑞(𝐺)>3+√5≈5.23606, then(1)either 𝑞(𝐺_𝑣1𝑣3)>𝑞(𝐺) or 𝑞(𝐺_𝑣1𝑣4)>𝑞(𝐺);(2)𝜇(𝐺_𝑣1𝑣3)≥𝜇(𝐺) and 𝜇(𝐺_𝑣1𝑣4)≥𝜇(𝐺).

Lemma 14 (see [18]). Suppose 𝑢 is a vertex of the bicyclic graph 𝐺 with 𝑑_𝐺(𝑢)≥2. Let 𝐺∶𝑢𝑣 be a graph obtained by attaching a pendant edge 𝑢𝑣 to 𝐺 at 𝑢. Suppose that a pendant edge 𝑣𝑤₁ and 𝑡 paths {𝑣𝑣₁𝑣^′₁,…,𝑣𝑣_𝑡𝑣^′_𝑡} of length 2 are attached to 𝐺∶𝑢𝑣 at 𝑣 to form 𝐿_1,𝑡. Let 𝑀_0,𝑡+1=𝐿_1,𝑡−𝑣𝑣₁−⋯−𝑣𝑣_𝑡+𝑢𝑣₁+⋯+𝑢𝑣_𝑡. Then we have(1)𝑞(𝑀_0,𝑡+1)>𝑞(𝐿_1,𝑡), (𝑡≥1);(2)𝜇(𝐿_1,𝑡)≤𝜇(𝑀_0,𝑡+1).

3. Main Results

Lemma 15. Let 𝐺₁,𝐺₂,…,𝐺₆ be the graphs as Figure 3. Then for 𝜇≥3, we have 𝑞(𝐺₁)>𝑞(𝐺_𝑖), (𝑖=2,3,…,6).

Figure 3 

𝐺1--𝐺6.

Proof. From Lemma 10, we have Φ𝑆𝐺₁=𝑥𝑥²−1𝑥⁴−3𝑥²+1^𝜇−3𝑥⁵−4𝑥³+3𝑥²−𝜇−3𝑥²−1𝑥³×𝑥−2𝑥₄−3𝑥²+1^𝜇−4𝑥⁵−4𝑥³+3𝑥²−𝑥𝑥⁴−3𝑥²+1^𝜇−3𝑥⁵−4𝑥³+3𝑥²−4𝑥²−1𝑥⁴−3𝑥²+1^𝜇−3×𝑥⁵−4𝑥³+3𝑥𝑥⁴−3𝑥²Φ𝑆𝐺+2₂=𝑥𝑥²−1𝑥⁴−3𝑥²+1^𝜇−4𝑥⁵−4𝑥³×𝑥+3𝑥₉−8𝑥⁷+19𝑥⁵−14𝑥³−𝑥+3𝑥𝜇−4²−1𝑥³−2𝑥𝑥⁴−3𝑥²+1^𝜇−5×𝑥⁵−4𝑥³+3𝑥𝑥⁹−8𝑥⁷+19𝑥⁵−14𝑥³𝑥+3𝑥−𝑥⁴−3𝑥²+1^𝜇−4𝑥⁵−4𝑥³×𝑥+3𝑥₉−8𝑥⁷+19𝑥⁵−14𝑥³+3𝑥−2𝑥²×𝑥−1₄−3𝑥²+1^𝜇−3×𝑥⁹−8𝑥⁷+19𝑥⁵−14𝑥³𝑥+3𝑥−2²−1𝑥⁴−3𝑥²+1^𝜇−4𝑥⁵−4𝑥³×𝑥+3𝑥₈−7𝑥⁶+14𝑥⁴−8𝑥²+1−2𝑥²×𝑥−1₄−3𝑥²+1^𝜇−4×𝑥⁹−8𝑥⁷+19𝑥⁵−14𝑥³𝑥+3𝑥−2²−1³𝑥⁴−3𝑥²+1^𝜇−4𝑥⁵−4𝑥³+3𝑥. (5) From (5), we have Φ𝑆𝐺₂−Φ𝑆𝐺₁=𝑥³𝑥⁴−3𝑥²+1^𝜇−5×−2+𝜇𝑥¹⁴+22−10𝜇𝑥¹²+−97+39𝜇𝑥¹⁰+221−75𝜇𝑥⁸+−278+74𝜇𝑥⁶+189−35𝜇𝑥⁴+−63+6𝜇𝑥²+8. (6) If 𝜇≥12, for 𝑥≥𝜌(𝑆(𝐺₁))≥Δ𝑆(𝐺₁)=𝜇+2, it is easy to prove that Φ(𝑆(𝐺₂))−Φ(𝑆(𝐺₁))>0. Hence, 𝜌(𝑆(𝐺₁))>𝜌(𝑆(𝐺₂)) for 𝜇≥12. When 𝜇=4,5,…,11, by direct calculation, we also get 𝜌(𝑆(𝐺₁))>𝜌(𝑆(𝐺₂)), respectively. So, 𝜌(𝑆(𝐺₁))>𝜌(𝑆(𝐺₂)) for 𝜇≥4. By Lemma 6, we know that 𝜌(𝑆(𝐺))=𝑞(𝐺). Hence, 𝑞(𝐺₁)>𝑞(𝐺₂)(𝜇≥4). By similar method, the result is as follows.

Theorem 16. If 𝐺∈𝐵_𝑛(2𝜇)(𝑛≥6), then 𝑞(𝐺)≤𝑞(𝐺₁), with equality if and only if 𝐺=𝐺₁.

Proof. Let 𝑋=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 be the Perron vector of 𝐺. From Lemma 12 and by direct calculations, we have, for 𝜇≥3, 𝑞(𝐺₁)>𝑞(𝐵(3,1,3))≈5.5615>3+√5. So, in the following, we only consider those graphs, which have signless Laplacian spectral radius greater than 𝑞(𝐺)>3+√5.
Choose 𝐺^∗∈𝐵_𝑛(2𝜇) such that 𝑞(𝐺^∗) is as large as possible. Then 𝐺^∗ consists of a subgraph 𝐻 which is one of graphs 𝐵(𝑝,1,𝑞), 𝐵(𝑝,𝑙,𝑞), and 𝑃(𝑝,𝑙,𝑞) (see Figures 1 and 2).
Let 𝑇 be a tree attached at some vertex, say, 𝑧, of 𝐻; |𝑉(𝑇)| is the number of vertices of 𝑇 including the vertex 𝑧. In the following, we prove that tree 𝑇 is formed by attaching at most one path of length 1 and other paths of length 2 at 𝑧.
Suppose 𝑃∶𝑣₀𝑣₁⋯𝑣_𝑘 is a pendant path of 𝐺^∗ and 𝑣_𝑘 is a pendant vertex. If 𝑘≥3, let 𝐻₁=𝐺^∗−𝑣₂𝑣₃+𝑣₀𝑣₃. From Corollary 8, we have 𝐻₁∈𝐵_𝑛(2𝜇) and 𝑞(𝐻₁)>𝑞(𝐺^∗), which is a contradiction.
For each vertex 𝑢∈𝑉(𝑇−𝑧), we prove that 𝑑(𝑢)≤2. Otherwise, there must exist some vertex 𝑢₀ of 𝑇−𝑧 such that 𝑑(𝑧,𝑢₀)=max{𝑑(𝑧,𝑣)∣𝑣∈𝑉(𝑇)−𝑧,𝑑(𝑣)≥3}. From the above proof, we have the pendant paths attached 𝑢₀ which have length of at most 2. Obviously, there exists an internal path between 𝑢₀ and some vertex 𝑤 of 𝐺^∗, denoted by 𝑃∶𝑢₀𝑤₁⋯𝑤_𝑚(𝑤_𝑚=𝑤). If 𝑚≥2, let 𝐻₂ be the graph obtained from 𝐺^∗−𝑢₀𝑤₁−𝑤₁𝑤₂ by amalgamating 𝑢₀, 𝑤₁, and 𝑤₂ to form a new vertex 𝑠₁ together with attaching a new pendant path 𝑠₁𝑠₂𝑠₃ of length 2 at 𝑠₁. From Corollary 5, we have 𝐻₂∈𝐵_𝑛(2𝜇) and 𝑞(𝐻₂)>𝑞(𝐺^∗), which is a contradiction. If 𝑚=1, by Lemma 14 and Corollary 3, we can get a new graph 𝐻₃ such that 𝐻₃∈𝐵_𝑛(2𝜇) and 𝑞(𝐻₃)>𝑞(𝐺^∗), which is a contradiction.
From the proof as above, we have the tree 𝑇 which is obtained by attaching some pendant paths of length 2 and at most one pendant path of length 1 at 𝑧.
From Corollary 2, we have all the pendant paths of length 2 in 𝐺^∗ which must be attached at the same vertex of 𝐻.
In the following, we prove that 𝐺^∗ is isomorphic to one of graphs 𝐺₁,𝐺₂,…,𝐺₆ (see Figure 3). We distinguish the following two cases:
Case  1 (𝐺^∗∈𝐵⁺_𝑛(2𝜇)). We prove that 𝐺^∗ is isomorphic to one of graphs 𝐺₁, 𝐺₂, and 𝐺₃.
Assume that there exists some cycle 𝐶_𝑝 of 𝐺^∗ with length of at least 4. From Corollary 5, we have each internal path of 𝐺^∗, which is not a triangle, has length 1. Note that all the pendant paths of length 2 in 𝐺^∗ must be attached at the same vertex, then there must exist edges 𝑣₁𝑣₂∈𝐸(𝐺^∗), 𝑣₁𝑣₃∈𝐸(𝐶_𝑝), and 𝑣₁𝑣₄∈𝐸(𝐶_𝑝) and 𝑑(𝑣₁)=3, 𝑑(𝑣₂)=1, 𝑑(𝑣₃)≥3, and 𝑑(𝑣₄)≥3. Let 𝐻₄(𝐻₅) be the graph obtained from 𝐺^∗−𝑣₁𝑣₃(𝐺^∗−𝑣₁𝑣₄) by amalgamating 𝑣₁ and 𝑣₃(𝑣₄) to form a new vertex 𝑦₁(𝑦₃) together with subdividing the edge 𝑦₁𝑣₂(𝑦₃𝑣₂) with a new vertex 𝑦₂(𝑦₄). From Lemma 13, we have 𝐻_𝑖∈𝐵⁺_𝑛(2𝜇)(𝑖=4,5) and either 𝑞(𝐻₄)>𝑞(𝐺^∗) or 𝑞(𝐻₅)>𝑞(𝐺^∗), which is a contradiction. Then for each cycle 𝐶_𝑔 of 𝐺^∗, we have 𝑔=3.
Assume that 𝑙≥4. If there exists an internal path 𝑃^∗∶𝑣_𝑖𝑣_𝑖+1⋯𝑣_𝑚(1≤𝑖<𝑚≤𝑙) with length greater than 1 in 𝐺^∗. Then, by Corollary 5, we can get a new graph 𝐻₆ such that 𝑞(𝐻₆)>𝑞(𝐺^∗) and 𝐻₆∈𝐵⁺_𝑛(2𝜇), which is a contradiction. Thus, 𝑑(𝑣_𝑖)≥3(𝑖=1,2,…,𝑙) and either 𝑑(𝑣₂)=3 or 𝑑(𝑣₃)=3. By Lemma 13, we can also get a new graph 𝐻₇ such that 𝑞(𝐻₇)>𝑞(𝐺^∗) and 𝐻₇∈𝐵⁺_𝑛(2𝜇), which is a contradiction. Hence, 𝑙≤3.
We distinguish the following three subcases:
Subcase  1.1 (𝑙=1). Then 𝐺^∗ is the graph obtained by attaching all the pendant paths of length 2 at the same vertex of 𝐺, where 𝐺 is one of graphs 𝐺₁,…,𝐺₅ (see Figure 4).
Assume that 𝐺=𝐺₂. If 𝑥_𝑢≥𝑥_𝑣, let 𝐻₈=𝐺^∗−𝑟𝑣−𝑠𝑣+𝑟𝑢+𝑠𝑢; if 𝑥_𝑣≥𝑥_𝑢, let 𝐻₉=𝐺^∗−𝑢𝑡+𝑡𝑣. Obviously, 𝐻_𝑖∈𝐵⁺_𝑛(2𝜇)(𝑖=8,9) and either 𝑞(𝐻₈)>𝑞(𝐺^∗) or 𝑞(𝐻₉)>𝑞(𝐺^∗) by Lemma 1, which is a contradiction. By similar reasoning, we have also 𝐺≠𝐺₃.
Subcase  1.2 (𝑙=2). Then 𝐺^∗ is the graph obtained by attaching all the pendant paths of length 2 at the same vertex of 𝐺, where 𝐺 is one of graphs 𝐺₆,…,𝐺₁₄ (see Figure 4).
Assume that 𝐺=𝐺₆. If 𝑥_𝑣1≥𝑥_𝑣2, let 𝐻₁₀=𝐺^∗−𝑣₂𝑢+𝑣₁𝑢; if 𝑥_𝑣2≥𝑥_𝑣1, let 𝐻₁₁=𝐺^∗−𝑣₁𝑟+𝑣₂𝑟. Obviously, 𝐻_𝑖∈𝐵⁺_𝑛(2𝜇)(𝑖=10,11) and either 𝑞(𝐻₁₀)>𝑞(𝐺^∗) or 𝑞(𝐻₁₁)>𝑞(𝐺^∗) by Lemma 1, which is a contradiction. By similar reasoning, we have also 𝐺≠𝐺_𝑗(𝑗=6,…,14).
Subcase  1.3 (𝑙=3). Then 𝐺^∗ is the graph obtained by attaching all the pendant paths of length 2 at the same vertex of 𝐺, where 𝐺 is one of graphs 𝐺₁₅,…,𝐺₂₀ (see Figure 4).
Assume that 𝐺=𝐺₁₅. If 𝑥_𝑣1≥𝑥_𝑣2, let 𝐻₁₂=𝐺^∗−𝑣₂𝑣₃+𝑣₁𝑣₃; if 𝑥_𝑣2≥𝑥_𝑣1, let 𝐻₁₃=𝐺^∗−𝑣₁𝑧₁+𝑣₂𝑧₁. Obviously, 𝐻_𝑖∈𝐵⁺_𝑛(2𝜇)(𝑖=12,13) and either 𝑞(𝐻₁₂)>𝑞(𝐺^∗) or 𝑞(𝐻₁₃)>𝑞(𝐺^∗) by Lemma 1, a contradiction. By similar reasoning, we have also 𝐺≠𝐺_𝑗(𝑗=15,…,20).
Thus, 𝐺 is isomorphic to one of the graphs 𝐺₁, 𝐺₄ and 𝐺₅. In the following, we prove that 𝐺^∗ is isomorphic to one of graphs 𝐺₁, 𝐺₂ and 𝐺₃.
Assume that 𝐺^∗ is obtained by attaching all the pendant paths of length 2 at vertex 𝑦₄ of 𝐺₁. If 𝑥_𝑣1≥𝑥_𝑦4, let 𝐻₁₄ be the graph obtained from 𝐺₁ by attaching 𝜇−3 pendant paths of length 2 at 𝑣₁. If 𝑥_𝑦4≥𝑥_𝑣1, let 𝐻₁₅=𝐺^∗−𝑣₁𝑦₃−𝑣₁𝑦₁−𝑣₁𝑦₂+𝑦₄𝑦₃+𝑦₄𝑦₁+𝑦₄𝑦₂. Obviously, 𝐻₁₄=𝐻₁₅=𝐺₁ and 𝑞(𝐺₁)>𝑞(𝐺^∗) by Lemma 1, a contradiction. Then 𝐺^∗=𝐺₁. By similar reasoning, the result follows.
Case  2 (𝐺^∗∈𝐵_𝑛⁺⁺(2𝜇)). By similar reasoning as that of Case  1, we have 𝐺^∗ is the graph obtained by attaching all the pendant paths of length 2 at the same vertex of 𝐺, where 𝐺 is one of graphs 𝐺₂₁,…,𝐺₂₄ (see Figure 4).
From Lemma 1, it is easy to prove that 𝐺≠𝐺₂₂ and all the pendant paths of length 2 are attached at the vertex of degree 3 of 𝐺₂₁ or of degree 4 of 𝐺_𝑖(𝑖=23,24). Thus, 𝐺^∗ is isomorphic to one of graphs 𝐺₄, 𝐺₅ and 𝐺₆ (see Figure 3).
So, 𝐺^∗ is isomorphic to one of graphs 𝐺₁,…,𝐺₆. From Lemma 15, we know 𝑞(𝐺₁)>𝑞(𝐺_𝑖),(𝑖=2,3,…,6). Thus, 𝐺^∗=𝐺₁.

Figure 4 

𝐺1--𝐺24.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

All authors completed the paper together. All authors read and approved the final paper.

Acknowledgments

This research is supported by NSFC (nos. 10871204, 61370147, and 61170309) and by Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020).