#### Abstract

In this paper, we study the properties and structure of the maximal-adjacency-spectrum unicyclic graphs with given maximum degree. We obtain some necessary conditions on the maximal-adjacency-spectrum unicyclic graphs in the set of unicyclic graphs with vertices and maximum degree and describe the structure of the maximal-adjacency-spectrum unicyclic graphs in the set. Besides, we also give a new upper bound on the adjacency spectral radius of unicyclic graphs, and this new upper bound is the best upper bound expressed by vertices and maximum degree from now on.

#### 1. Introduction

The spectral theory of graphs was established in the 1940s and 1950s. It is a branch of mathematics that is widely applied, and itis a powerful tool for solving problems in discrete mathematics. Many of the early results were related to the relationship between the spectrum and the structural properties of a graphs [1–4]. The spectral theory of graphs has been widely used in quantum theory, chemistry, physics, computer science, the theory of communication networks, and information science. Along with the continuous research of the spectral theory of graphs, applications of the spectral theory of graphs have also been found in the fields of electrical networks and vibration theory [5, 6].

Not only has the spectral theory of graphs pushed forward and enriched research into combinatorics and graph theory but also it has been widely used in quantum theory, chemistry, physics, computer science, the theory of communication networks, and information science. The wide range of application of the spectral theory of graphs has led to the spectral theory of graphs becoming a very active field of research over the last thirty to forty years, and large numbers of results are continuously emerging.

There are many results on the adjacency spectral radius for different classes of graphs. Guo et al. [7] have studied the largest and the second largest spectral radius of trees with vertices and diameter . Guo and Shao [8] have studied the first spectral radii of graphs with vertices and diameter . Petrovic et al. [9, 10] have studied the spectral radius of unicyclic and bicyclic graphs with vertices and pendant vertices. Guo et al. [11, 12] have studied the spectral radius of unicyclic and bicyclic graphs with vertices and diameter *d*.

Let be a connected graph with edge set and vertex set . In this paper, we denote by a edge of , where . Denote the maximum degree of vertex of by . For convenience, we shall sometimes denote simply by . Denote the degree of vertex by . Denote by the set which consists of the vertices adjacent to vertex in . Denote by the shortest distance between vertex and vertex . Denote by the adjacency spectral radius of . If is a connected graph with vertices, where is the vertex set of , is the edge set of , and , then is called a unicyclic graph. We denote the set which consists of the unicyclic graphs with vertices and maximum degree by .

In this paper, we study further the properties and structure of the maximal-adjacency-spectrum unicyclic graphs in the set of unicyclic graphs with vertices and the maximum degree , where . Besides, we study the new upper bound on the adjacency spectral radius of unicyclic with vertices and the maximum degree .

Suppose that , then we obtain some necessary conditions about that is maximal-adjacency-spectrum unicyclic graph in using the following theorems.

Theorem 1. *Suppose that is a maximal-adjacency-spectrum unicyclic graph in , if is the Perron vector of , then the degree of the vertices that corresponds to all largest components in the component of is .*

Theorem 2. *Suppose that is a maximal-adjacency-spectrum unicyclic graph in and the only circle in is , is the Perron vector of ; if the component which corresponds to the vertex in satisfies that , then .*

Theorem 3. *Suppose that is a maximal-adjacency-spectrum unicyclic graph in , if the only circle in is , and there exists a nonfull internal vertex in the set of , then the number of all the nonfull internal vertices of is one.*

Theorem 4. *Suppose that is a maximal-adjacency-spectrum unicyclic graph in , let be the only circle in and , if there is no nonfull internal vertex in the set of , then both the following propositions are established:*(1)*The number of the nonfull internal vertex in is at most 2.*(2)*When the number of nonfull internal vertex in is 2, then there is at least one vertex with degree 2 in the two nonfull internal vertices in .*

Theorem 5. *Suppose that is a maximal-adjacency-spectrum unicyclic graph in . If the only circle in is , there is no nonfull internal vertex in the set of , and the number of the nonfull internal vertex in the set of is equal to or greater than 2, then the length of the circle is 3, that is, .*

Theorem 6. *Suppose that is a maximal-adjacency-spectrum unicyclic graph in , if is the only circle in , then .*

Theorem 7. *Suppose that is a maximal-adjacency-spectrum unicyclic graph in , and let be the vertex that corresponds to a maximum component in the component of the Perron vector of . is the rooted unicyclic graph with root node , let be the only circle in , and . Let be the shortest distance between vertex and vertex in , denote that = { is the leaf node of , and the shortest path from to neither pass nor pass }, = { is the leaf node of , and the shortest path from to either pass or pass }, then the following propositions are established:*(1)*If , then .*(2)*If , then for the arbitrary leaf node in , we all have .* *If , then for the arbitrary leaf node in , we all have .*(3)*If , then for the arbitrary leaves in , we all have .*(4)*If , then .*(5)*If and there exists a vertex with degree 2 in , then is the rooted unicyclic graph with 3 levels, and for , we all have .*

Theorem 8. *Suppose that is a maximal-adjacency-spectrum unicyclic graph in , is the Perron vector of , is the vertex that corresponds to a maximum component in the component of the Perron vector of . Let be the rooted unicyclic graph with root node , then is an almost full-degree unicyclic graph with root node .*

Theorem 9. *Suppose that is a maximal-adjacency-spectrum unicyclic graph in , is the Perron vector of , is the vertex that corresponds to a maximum component in the component of the Perron vector of , and is the rooted unicyclic graph with root node , has only one nonfull internal vertex . Suppose that is the only one circle in , and , denote that = { is the leaf node of , and the shortest path from to neither pass nor pass }, = { is the leaf node of , and the shortest path from to either pass or pass }. Assume that , that is, , then the following propositions are established:*(1)*If the distances from all leaves of to are all equal, then there exists a leaf node which makes is a pendant edge.*(2)*If there are two leaves in which make and , then either there exists a leaf node , which makes is a pendant edge or and .*(3)*If , then there exists a leaf node , which makes is a pendant edge.*

Finally, we obtain the main result of this paper in the following theorem.

Theorem 10. *Suppose that , and is a rooted unicyclic graph with root vertex which is the vertex that corresponds to a maximum component in the component of the Perron vector of , then is a maximal-adjacency-spectrum unicyclic graph in if and only if .*

That is, we describe the structure of the maximal-adjacency-spectrum unicyclic graphs in . In addition, we give a new upper bound on the adjacency spectral radius of unicyclic graphs on the basis of Theorem 10.

In the following discussion of this paper, we assume that the maximum degree of unicyclic graphs satisfies .

#### 2. Preliminaries

##### 2.1. Some Basic Concepts

A rooted unicyclic graph is a simple nonlinear structure. Figure 1 shows a rooted unicyclic graph with root node .

We can divide a rooted unicyclic graph into levels according to the following principles.

The root nodes are in the first level. For instance, in the rooted unicyclic graph with root node shown in Figure 1, is in the first level. In a rooted unicyclic graph, the level which arbitrary vertex is defined as the shortest distance from the vertex to the root node adds 1; for instance, in the rooted unicyclic graph with root node shown in Figure 1, the vertices are in the second level, the vertices are in the third level, and the vertices are in the fourth level. The levels of a rooted unicyclic graph are defined as the maximum of the levels of all the vertices in the rooted unicyclic graph. For instance, the level of the rooted unicyclic graph shown in Figure 1 is 4.

Assume that is a rooted unicyclic graph, is the root node of and is an arbitrary vertex which is not equal to in . Suppose that the shortest path from to is , where , then we call is a father vertex of . For instance, in the rooted unicyclic graph with root node shown in Figure 1, the shortest path from vertex to vertex is , is a father vertex of . The shortest paths from vertex to vertex are , respectively; therefore, and are both father vertices of .

Let be a rooted unicyclic graph, if and are two different vertices of , and have the same father vertex, then is called a brother of . If and are two different vertices in the same level in , and the father vertex of is not equal to the father vertex of , then is called a cousin of . For instance, in the rooted unicyclic graph with root node shown in Figure 1, vertex , , and have the same father vertex ; therefore, is a brother of . Besides, vertices are two vertices in the same level in , and the father vertex of is not equal to the father vertex of ; therefore, is a cousin of .

The vertices (except ) on the shortest path that connects the root node to vertex are called the direct ancestor of the vertex . For instance, in the rooted unicyclic graph with root node shown in Figure 1, the shortest path from the root node to the vertex is ; therefore, and are both the direct ancestors of .

Assume that are two different vertices in the rooted unicyclic graph , the level of the vertex in is and the level of the vertex in is , where and is not the direct ancestor of , then we call is a collateral ancestor of . For instance, in the rooted unicyclic graph with root node shown in Figure 1, the level of vertex in is 3, the level of vertex in is 2, and is not the direct ancestor of ; therefore, vertex is the collateral ancestor of vertex .

Suppose that is a rooted unicyclic graph, if vertex and vertex are both direct ancestors of vertex , and the level of vertex in is larger than the level of vertex in , then the generation of vertex to is closer than the generation of vertex to vertex . For instance, in the rooted unicyclic graph with root node shown in Figure 1, the shortest path from vertex to vertex is , vertices are both direct ancestors of , the root node is in the first level, and vertex is in the second level; therefore, the generation of vertex to is closer than the generation of vertex to vertex .

If , and for any number which satisfies , we all have that is a direct ancestor of , then is called a common direct ancestor of . For instance, in the rooted unicyclic graph with root node shown in Figure 1, is both the direct ancestor of and the direct ancestor of ; therefore, is a common direct ancestor of .

If , is a common direct ancestor of , and which is the arbitrary direct ancestor of satisfies that is either equal to or the direct ancestor of , then is called the common direct ancestor of of the nearest generation. For instance, in the rooted unicyclic graph with root node shown in Figure 1, is the common direct ancestor of of the nearest generation.

Let be two cousins of , if the common direct ancestor of and of the nearest generation is the common direct ancestor of and of the nearest generation, then is the cousin of with generation closer than . For instance, in the rooted unicyclic graph with root node shown in Figure 1, are two cousins of , and the common direct ancestor of and of the nearest generation is , the common direct ancestor of and of the nearest generation is , and is the ancestor of ; therefore, is the cousin of generation closer than .

In order to give the main results of this paper, we introduce some basic definitions and lemmas.

##### 2.2. Some Definitions

*Definition 1. *Suppose that , if for any all have that establish, then is called a maximal-adjacency-spectrum unicyclic graph in .

*Definition 2. *Suppose that and are vertices different from each other in graph , if , and for any natural number which satisfies all have , then a path of graph is called an internal path of graph .

*Definition 3. *Suppose that is a unicyclic graph, and the degree of is , is called a nonfull vertex of , which means satisfies .

*Definition 4. *Suppose that is a simple connected graph with vertices, the vertices in with degree 1 are called the pendant point of , or call that vertex in the leaf node of ; for convenience, the leaf node of is sometimes simply called leaf node of . The edge associated with the pendant point is called pendant edge.

*Definition 5. *Suppose that is a unicyclic graph, and the maximum degree of is , if satisfies or , where is shown in Figure 2, and is the root node of . Or is a rooted unicyclic graph with levels more than two, and satisfies the following properties:(1)The vertex in the first level is , and is the root node of ; the vertices in the second level from left to right are .(2)Suppose that the only circle in is , and .(3)The internal vertices of are all full-degree vertices.(4)The distance from all the leaf node nodes of to is equal.Then, is called a completely full degree unicyclic graph.

*Definition 6. *Suppose is a unicyclic graph, and the maximum degree of is , if satisfies or that , where is shown in Figure 2, and is the root node of . Or is a completely full degree unicyclic graph with levels more than two. Or is a rooted unicyclic graph obtained from another completely full degree unicyclic graph with levels more than two by deleting some right leaf nodes. Then, is called an almost completely full degree unicyclic graph. We denote the almost completely full degree unicyclic graph with vertices and maximum degree by .

*Definition 7. *Suppose that is a unicyclic graph, and the maximum of is , if satisfies or , where is shown in Figure 2, and is the root node of . Or , where (where is the number which satisfies that ), , and is the root node of . or is a rooted unicyclic graph with levels, and satisfies the following properties:(1).(2)The vertex in the first level is , and is the root node of ; the vertices in the second level from left to right are .(3)Suppose that the only circle in is , and .(4)There is at most one nonfull internal vertex in .(5)When there is only one nonfull internal vertex in , this nonfull internal vertex is in the level of .Then, is an almost full degree unicyclic graph with root node .

From the definitions above, we know that completely full degree unicyclic graph is the special situation of almost completely full degree unicyclic graph. completely full degree unicyclic graph and almost completely full degree unicyclic graph are both special situations of almost full degree unicyclic graph. For convenience, in this paper, we denote completely full degree unicyclic graph, almost completely full degree unicyclic graph, almost full degree unicyclic graph by completely full-degree unicyclic graph, almost completely full-degree unicyclic graph, and almost full-degree unicyclic graph, respectively.

For instance, suppose that the unicyclic graph shown in Figure 3(a) is a rooted unicyclic graph with root node , then the rooted unicyclic graph shown in Figure 3(a) is a completely full 3 degree unicyclic graph with levels 3, maximum degree 3, and root node . Suppose that the root node of the unicyclic graph shown in Figure 3(b) is , then the rooted unicyclic graph shown in Figure 3(b) is an almost completely full 4 degree unicyclic graph with levels 4, maximum degree 4, and root node . Suppose that the root node of the unicyclic graph shown in Figure 3(c) is , then the rooted unicyclic graph shown in Figure 3(c) is an almost full 4 degree unicyclic graph with levels 4, maximum degree 4, and root node .

In order to give the main results of this paper, we give some lemmas.

**(a)**

**(b)**

**(c)**

##### 2.3. Some Lemmas

Now, we give some lemmas which we use to proof the main results.

Lemma 1 (see [13]). *Suppose that is a simple connected graph with vertices and maximum , is the Perron vector of , correspond to vertices , respectively, and . If , let , if is still a simple connected graph, then .*

Lemma 2 (see [13]). *Suppose that is a simple connected graph with vertices and maximum , is the Perron vector of , correspond to vertices , respectively, and . Suppose that . Let , and are shown in Figure 4. If is still a simple connected graph, then .*

Lemma 3 (see [13]). *Suppose that is a simple connected graph with vertices and maximum , is the Perron vector of , and suppose that correspond to the four different vertices , respectively, where and . Let , and are shown in Figure 5; if is still a simple connected graph, then .*

Lemma 4 (see [14]). *Suppose that is a simple connected graph, and is an internal path of . If , where , then .*

Lemma 5 (see [13]). *Suppose that is a connected graph, if , let , then .*

Lemma 6. *Assume that , the edge set of is , and is the only circle in . Suppose that , where , and for any natural number which satisfies , we all have , denote . Suppose that is the Perron vector of , (where ) is the set which consists of all the vertices whose components of Perron vector are equal to in , if for any arbitrary natural number which satisfies , we all have , then there exists which makes that holds.*

*Proof. *Since for any arbitrary natural number which satisfies , we all have , we can get . Without loss of generality, we assume that have clockwise arrangement in , as shown in Figure 6, and denote .

Since for any natural number which satisfies , we all have . We know that there exist two vertices in and exist two numbers and that satisfy both , and , which make is in clockwise arrangement between and , is adjacent to , and is in clockwise arrangement between and , is adjacent to , and . From the definition of , we know . Let , then it is easy to know . From and , by Lemma 3, we have ; therefore, Lemma 6 holds.

Lemma 7. *Assume that are four different vertices of , and . If is the Perron vector of , correspond to vertices , respectively, and . Let , if is still a simple connected graph, then .*

*Proof. *Since is the Perron vector of , we can imply that , where is the adjacency matrix of . For , and is a simple connected graph; hence, we get .

Therefore, Lemma 7 holds.

#### 3. The Properties and Structure of the Maximal-Adjacency-Spectrum Unicyclic Graphs in

##### 3.1. The Properties of the Maximal-Adjacency-Spectrum Unicyclic Graphs in

First, we give the properties of the Perron vector of the maximal-adjacency-spectrum unicyclic graphs in , as in the following theorems:

Theorem 11. *Suppose that is a maximal-adjacency-spectrum unicyclic graph in , if is the Perron vector of , then the degree of the vertices that corresponds to all largest components in the component of is .*

*Proof. *Assume that the proposition is not established, we suppose that there exists a vertex in such that and . Denote , it is obvious that ; hence, . We choose a vertex with degree in , suppose that . For is a unicyclic graph, it is easy to know that there are vertices in the set of such that is still a simple connected graph. Denote , then it is easy to know . Again by , by Lemma 2, we get , and this implies a contradiction with is a maximal-adjacency-spectrum unicyclic graph in ; hence, the hypothesis is not established; therefore, Theorem 11 holds.

Theorem 12. *Suppose that is a maximal-adjacency-spectrum unicyclic graph in , and the only circle in is , is the Perron vector of , if the component which corresponds to the vertex in satisfies that , then .*

*Proof. *Assume that the proposition is not established, we suppose that there exists a vertex in such that and . From and Theorem 11, we get . For and is a unicyclic graph, we know that there exists a path whose length is in , which makes , are difference with each other, and is the leaf node.

First, for , we have . Otherwise, suppose there exists a vertex in such that . Now according to the value of , we discuss the following two cases: Case 1. . Since , we have ; thus in the set of , there exists a vertex which satisfies such that is still a simple connected graph. Denote , then it is easy to know . By Lemma 1, we get , and this implies a contradiction with is a maximal-adjacency-spectrum unicyclic graph in . Case 2. . In this case, it is easy to know that in the set of , there exist vertices such that is still a simple connected graph. Denote , then it is easy to know . From , by Lemma 2, we can get , and this implies a contradiction with is a maximal-adjacency-spectrum unicyclic graph in .From the above discussion of the two cases, we know that the hypothesis is not established. Hence, for , we all have that holds.

Second, denote by , suppose that is the vertex which is nearest to in . Choose two vertices different from in , then , and . Thus, we have that there exists a natural number which satisfies such that at least one of the following two inequality groups: ① and ② holds. Without loss of generality, we assume that there exists a natural number which satisfies such that and . Then, let , then it is easy to know . By Lemma 3, we have that holds, and this implies a contradiction with is a maximal-adjacency-spectrum unicyclic graph in . Therefore, Theorem 12 holds.

Lemma 8. *Assume that , is the Perron vector of , and is the only circle in , , suppose that corresponds to the vertices , respectively. If , and , then there must exist such that .*

*Proof. *Assume that the proposition is not established, then is a maximal-adjacency-spectrum unicyclic graph in . From Theorem 12, we can get ; again by Theorem 11, we get ; hence, for any natural number which satisfies , and we all have . Besides, from Theorem 12, we get that for , we all have .

Without loss of generality, we assume that have clockwise arrangement in . Since , choose , then we have . For , we can know are different from each other. From and , we know that ; again from , we can get . Let , then it is easy to know . From and , by Lemma 3, we have that holds, and this implies a contradiction with is a maximal-adjacency-spectrum unicyclic graph in . Therefore, Lemma 8 holds.

Lemma 9. *Assume that , where , is the circle of , , and for any natural number which satisfies , we all have . If there exists a natural number which satisfies such that , then there exists such that .*

*Proof. *Assume that the proposition is not established, then is a maximal-adjacency-spectrum unicyclic graph in .

Form Theorems 11 and 12, we know that there exists a natural number which satisfies such that and . Now, we discuss the following two cases: Case 1. There are at least two vertices with the degree not less than 3 in . Denote , for any natural number which satisfies , we all have ; hence, for any natural number which satisfies , we all have . And for there exists a natural number which satisfies such that and there are at least two vertices with the degree not less than 3 in , then we know that there exist two natural numbers and which satisfy and such that is an internal path. Let , where . It is easy to know , and by Lemma 4, we have that holds. Suppose that is an arbitrary leaf node of . Let , then it is easy to know , and by Lemma 5, we have that holds, and thus, . Therefore, this implies a contradiction with is a maximal-adjacency-spectrum unicyclic graph in . Case 2. There is only one vertex with degree not less than 3 in . In this case, it is easy to know that the only one vertex with degree not less than 3 must be . Suppose that , and have clockwise arrangement in , as shown in Figure 7. From Theorems 11 and 12, we get . From , we know that are different. And for , it is easy to prove that holds. Otherwise, , let , then it is easy to know . From and , by Lemma 3, we have that holds, and this implies a contradiction with is a maximal-adjacency-spectrum unicyclic graph in ; hence, . Let , then it is easy to know , and by Lemma 1, we have that holds, and this implies a contradiction with is a maximal-adjacency-spectrum unicyclic graph in .By consideration of the above two cases, we know that the hypothesis is not established. Therefore, Lemma 9 holds.

Lemma 10. *Assume that is a maximal-adjacency-spectrum unicyclic graph in , and is the Perron vector of . Then, for any leaf node and any vertex which is not the leaf node in , we all have that holds, where correspond to the vertices , respectively.*

*Proof. *Suppose that is a maximal-adjacency-spectrum unicyclic graph in , and is the Perron vector of . Let be any one of the vertices whose component is equal to in , then from Theorem 11, we know . Then, for any nonleaf node which is not in , and for the arbitrary leaf node in , we all have that holds. Otherwise, there exists a vertex which is not the vertex and is not a leaf node in , which makes . For is not the leaf node, hence , thus there exists a vertex in the set of such that still belong to . Let , then by Lemma 1, we have that holds, this implies a contradiction with is a maximal-adjacency-spectrum unicyclic graph in . Again since , then ; thus, for any vertex which is not the leaf node in , and for any leaf node in , we all have that holds.

For the properties of the nonfull internal vertices of the maximal-adjacency-spectrum unicyclic graph in , we have the following theorems.

Theorem 13. *Suppose that is a maximal-adjacency-spectrum unicyclic graph in , if the only circle in is , and there exists a nonfull internal vertex in the set of , then the number of all the nonfull internal vertices of is one.*

*Proof. *Assume that the proposition is not established, then suppose that is a maximal-adjacency-spectrum unicyclic graph in , the only circle in is , there exists a nonfull internal vertex in the set of , and the number of the nonfull internal vertex in is more than 1. Without loss of generality, assume that are two nonfull internal vertices in , where .

Let be the Perron vector of , correspond to the vertices , respectively. Now according to whether and adjacent or not , we discuss the following two cases: Case 1. and adjacent. For case 1, we discuss the following two subcases again according to the value of and . Subcase 1. . Since is the nonfull internal vertex in , then , choose . Let , then it is easy to know . By Lemma 1, we have that holds, and this implies a contradiction with is a maximal-adjacency-spectrum unicyclic graph in . Subcase 2. . Since is the nonfull internal vertex in , then , choose . Let , then it is easy to know . By Lemma 1, we have that holds, and this implies a contradiction with is a maximal-adjacency-spectrum unicyclic graph in . Case 2. and are not adjacent. Since are the nonfull internal vertices in , we have that and hold at the same time. For case 2, we discuss the following two subcases again according to the value of and . Subcase 1. . Since and is a unicyclic graph, we know that there exists such that is still a simple connected graph. Denote , then it is easy to know ; by Lemma 1, we have that holds, and this implies a contradiction with is a maximal-adjacency-spectrum unicyclic graph in . Subcase 2. . For and is a unicyclic graph, we know that there exists such that is still a simple connected graph. Denote , then it is easy to know ; by Lemma 1, we have that holds, and this implies a contradiction with is a maximal-adjacency-spectrum unicyclic graph in .Hence, from the discussions of case 1 and case 2, we can know that the hypothesis is not established; therefore, Theorem 13 holds.

Theorem 14. *Suppose that is a maximal-adjacency-spectrum unicyclic graph in , let be the only circle in , and , if there is no nonfull internal vertex in the set of , then both the following propositions are established:*(1)*The number of the nonfull internal vertex in is at most 2.*(2)*When the number of nonfull internal vertex in is 2, then there is at least one vertex with degree 2 in the two nonfull internal vertices in .*

*Proof. *Let be the Perron vector of .(1)Suppose that a vertex in satisfies , then from Theorem 12, we get