Abstract
The first four smallest values of the spectral radius among all connected graphs with maximum clique size are obtained.
1. Introduction
Let be a simple connected graph with vertex set and edge set . Its adjacency matrix is defined as matrix , where if is adjacent to and , otherwise. Denote by or the degree of the vertex . It is well known that is a real symmetric matrix. Hence, the eigenvalues of can be ordered as respectively. The largest eigenvalue of is called the spectral radius of , denoted by . It is easy to see that if is connected, then is nonnegative irreducible matrix. By the Perron-Frobenius theory, has multiplicity one and exists a unique positive unit eigenvector corresponding to . We refer to such an eigenvector corresponding to as the Perron vector of .
Denote by and the path and the cycle on vertices, respectively. The characteristic polynomial of is , which is denoted by or . Let be an eigenvector of corresponding to . It will be convenient to associate with a labelling of in which vertex is labelled (or ). Such labellings are sometimes called “valuation” [1].
The investigation on the spectral radius of graphs is an important topic in the theory of graph spectra. The recent developments on this topic also involve the problem concerning graphs with maximal or minimal spectral radius, signless Laplacian spectral radius, and Laplacian spectral radius, of a given class of graphs, respectively. The spectral radius of a graph plays an important role in modeling virus propagation in networks [2]. It has been shown that the smaller the spectral radius, the larger the robustness of a network against the spread of viruses [3]. In [4], the first three smallest values of the Laplacian spectral radii among all connected graphs with maximum clique size are given. And, in [5], it is shown that among all connected graphs with maximum clique size the minimum value of the spectral radius is attained for a kite graph , where is a graph on vertices obtained from the path and the complete graph by adding an edge between an end vertex of and a vertex of (shown in Figure 1). Furthermore, in this paper, the first four smallest values of the spectral radius are obtained among all connected graphs with maximum clique size .
Let be the set of all connected graphs of order with a maximum clique size , where . It is easy to see that . By direct calculation, we have . If , then, from the Perron-Frobenius theorem, the first smallest values of the spectral radius of are (), respectively, where is the graph obtained from by adding () edges. So in the following, we consider that .
2. Preliminaries
In order to complete the proof of our main result, we need the following lemmas.
Lemma 1 (see [6]). Let be a vertex of the graph . Then the inequalities hold. If is connected, then .
For the spectral radius of a graph, by the well-known Perron-Frobenius theory, we have the following.
Lemma 2. Let be a connected graph and a proper subgraph of . Then .
Lemma 3 (see [6, 7]). Let be a graph on vertices, then The equality holds if and only if is a regular graph.
Let be a vertex of a graph and suppose that two new paths and of lengths and () are attached to at , respectively, to form a new graph (shown in Figure 2), where and are distinct. Let We call that is obtained from by grafting an edge (see Figure 2).
Lemma 4 (see [8, 9]). Let be a connected graph on vertices and is a vertex of . Let and () be the graphs as defined above. Then .
Let be a vertex of the graph and the set of vertices adjacent to .
Lemma 5 (see [10, 11]). Let be a connected graph, and let be two vertices of . Suppose that () and is the Perron vector of , where corresponds to the vertex (). Let be the graph obtained from by deleting the edges and adding the edges (). If , then .
Lemma 6 (see [12]). Let be a vertex of , let be the collection of circuits containing , and let denote the set of vertices in the circuit . Then the characteristic polynomial satisfies where the first summation extends over those vertices adjacent to , and the second summation extends over all .
An internal path of a graph is a sequence of vertices with such that(1)the vertices in the sequence are distinct (except possibly );(2) is adjacent to , ();(3)the vertex degrees satisfy (unless ) and .
Let be the tree on vertices obtained from by attaching two new pendant edges to each end vertex of , respectively.
Lemma 7 (see [13]). Suppose that is a connected graph and is an edge on an internal path of . Let be the graph obtained from by subdivision of the edge . Then .
3. Main Results
Let be the graph obtained from and a path by joining a vertex of and a nonpendant vertex, say, , of by a path with length 2 and let be the graph obtained from by attaching two pendant edges at two different vertices of (see Figure 3).
Lemma 8. Let and be the graphs defined as above (see Figure 3). If , then .
Proof. For , by direct computations, we have . In the following, we suppose that . From Lemma 6, we have
By direct calculation, we have
From Lemmas 1 and 3, we have and . Then from (7) we have .
Let be the graph obtained from the kite graph (see Figure 1) and an isolated vertex by adding an edge () (see Figure 4). It is easy to see that and .
Let (see Figure 5).
Lemma 9. Let be the graphs defined as above (see Figure 4). Then
Proof. Clearly, , . From Lemma 4, we have For , from Lemma 2, we have .
Let , let , and let be the graph obtained from and an isolated vertex by adding an edge between some vertex of and the isolated vertex (see Figure 6).
Theorem 10. Among all connected graphs on vertices with maximum clique size and , the first four smallest spectral radii are exactly obtained for , , , , and , respectively.
Proof. Let be a connected graph with maximum clique size and vertices. From Lemma 9, we have . Thus, we only need to prove that if , , , , . If is a tree, note that , , , , then, from Lemma 4, we have . If contains some cycle as a subgraph, then, from Lemmas 2 and 7, we have .
Lemma 11. Let , , and be the graphs defined as above (see Figures 4, 5, and 6). Then
Proof. For , by direct calculation, we have . If , from Lemmas 2 and 7, we have . From Lemma 6, we have Then we have For , we have Note that from Lemma 2, and . Then, we have Thus, . By similar method, we have for
Let be the graph obtained from by attaching two pendant edges at some vertex of ; let be the graph obtained from and by adding two edges between two vertices of and two end vertices of (see Figure 7).
Theorem 12. Among all connected graphs on vertices with maximum clique size and , the first four smallest spectral radii are exactly obtained for , , , , respectively.
Proof. Let be a connected graph with maximum clique size and vertices. From Lemmas 2 and 7, we have
Thus, we only need to prove that if , , , .
We distinguish the following three cases.
Case 1. If there exist at least two vertices outside of that are adjacent to some vertices of , then we have that contains either () or () as a proper subgraph. If contains () as a proper subgraph, from Lemmas 2 and 7, we have
If contains () as a proper subgraph, from Lemmas 2 and 7, we have
Case 2. Suppose that there exists a vertex, say, , which does not belong to , such that is adjacent to at least two vertices of . Then contains as a proper subgraph, where is obtained from by adding an edge between two disjoint vertices. From Lemmas 2 and 7, we have
Case 3. Suppose that there uniquely exists a vertex which does not belong to such that is adjacent to a vertex of . We distinguish the following two cases.
Subcase 1. Suppose that is a tree. If there exist two vertices such that and , then, from Lemmas 2, 4, and 7, we have . If there exists only one vertex such that , then, from Lemmas 2, 7, and 11, we have . If there exists exactly one vertex such that , note that , , , then from Lemmas 2 and 7 we have .
Subcase 2. Suppose that contains cycle as a subgraph. If , then, from Lemmas 2, 7 and 11, we have or . If , then, from Lemma 2, we can construct a graph from by deleting vertices such that , where is the graph obtained from and a cycle by joining a vertex of and a vertex of with a path and (see Figure 7). Suppose that is labelled satisfying , (), , and . Then, from Lemmas 2 and 7, we have . Thus, we have .
Lemma 13. Let and be the graphs defined as above (see Figures 4 and 5). Then ().
Proof. Let be the Perron vector of , where corresponds to . It is easy to prove that . From , we have From Lemma 2, for we have . Then So, .
Let () be the graph as shown in Figure 8.
Theorem 14. Among all connected graphs on vertices with maximum clique size and , the first four smallest spectral radii are exactly obtained for , , , , respectively.
Proof. Let be a connected graph with maximum clique size and vertices. Suppose that is a maximum clique of . From Lemmas 2, 4, and 13, we have
Thus, we only need to prove that if , , , . We distinguish the following three cases.
Case 1. If there exist at least two vertices outside of that are adjacent to some vertices of , then contains either or as a proper subgraph. If contains as a proper subgraph, from Lemmas 2, 7, and 8, we have
If contains as a proper subgraph, from Lemmas 2, 5, 7, and 8, we have
Case 2. Suppose that there exists a vertex, say, , which does not belong to , such that is adjacent to at least two vertices of . From Lemmas 2, 7, and 8, we have
Case 3. Suppose that there uniquely exists a vertex which does not belong to such that is adjacent to a vertex of . If is a tree, note that , , , then, from Lemmas 2, 4, and 7, we have . Suppose that contains cycle as a subgraph. If , note that , then, from Lemmas 2 and 7, we have , where . If , then by the similar reasoning as that of Subcase 2 of Case 3 of Theorem 12, we have .
Lemma 15. Let and be the graphs defined as above (see Figure 7). Then
Proof. Let be the Perron vector of , where corresponds to . From , we have From above equations, we have Let Then For and , we have Note that . From (30) and (31), we have which is the largest root of equation . Similarly, we have which is the largest root of equation Then we have, for , Thus, we have .
Theorem 16. Let be a graph on vertices with maximum clique size and . Let , , , and be the graphs defined as above (see Figures 1, 3 and 7). The first four smallest spectral radii are obtained for , , , , respectively.
Proof. From Lemmas 2, 5, 8, and 15, we have
Thus, we only need to prove that, for , , , and , . We distinguish the following two cases.
Case 1. Suppose that there exists exactly one vertex outside of that is adjacent to at least two vertices of . Then contains (see Figure 8) as a subgraph. From Lemmas 2 and 7, we have .
Case 2. Suppose that the two vertices outside of that are all adjacent to some vertices of . Note that , , . Then contains one of graphs and as a subgraph, where is obtained from by adding an edge between two pendant vertices. From Lemma 5, we have . From Lemmas 2 and 7, .
Let be the graph obtained from and an isolated vertex by adding an edge between a pendant vertex of and the isolated vertex; let and be the graphs as shown in Figure 9.
Lemma 17. Let and be the graphs defined as above (see Figure 9). Then
Proof. Let be the Perron vector of , where corresponds to . It is easy to see that . From , we have From above equations, we have Then for , we have The result follows.
Lemma 18. Let and be the graphs defined as above (see Figure 9). Then
Proof. For , by direct calculation, we have . In the following, we suppose that . Then, from Lemmas 2 and 3, we have . Let be the Perron vector of , where corresponds to . From , we have From above equations, we have for , Then, from Lemma 5, we have .
Let be the graph obtained from and an isolated vertex by adding an edge between and the isolated vertex; let be the graph obtained from and an isolated vertex by adding an edge between and the isolated vertex; let be the graph obtained from and an isolated vertex by adding an edge between one pendant vertex and the isolated vertex; and let be the graph obtained from and an isolated vertex by adding an edge between and the isolated vertex (see Figure 10).
Theorem 19. Let , , , and be the graphs defined as above (see Figures 1, 4, 5, and 9). Among all connected graphs on vertices with maximum clique size and (), the first four smallest spectral radii are obtained for , , , and , respectively.
Proof. From Lemmas 2, 4, and 17, we have
Thus, we only need to prove that if , , , and . We distinguish the following four cases.
Case 1. There exists exactly one vertex outside of that is adjacent to only one vertex of . Then must be one of graphs , , and .
Case 2. There exists one vertex outside of that is adjacent to at least two vertices of . Then contains (see Figure 8) as a proper subgraph. From Lemmas 2 and 7, we have .
Case 3. If there exactly exist two vertices outside of that are adjacent to some vertices of , then contains or (see Figures 9 and 10) as a subgraph. If contains as a subgraph, then, from Lemmas 2 and 5, we have . If contains as a subgraph, note that , then, from Lemma 2, we have .
Case 4. If there exist three vertices outside of that are adjacent to some vertices of , then contains one of graphs , , and (see Figures 9 and 10) as a subgraph. From Lemmas 5 and 18, we have . Then, from Lemma 2, we have .
Lemma 20. Let and be the graphs defined as above (see Figures 4 and 5). Then
Proof. From Lemma 6, we have Then, we have For (), we have From Lemma 2, we have and . Thus, for (), we have . Then , ().
Lemma 21. Let and be the graphs defined as above (see Figures 3 and 4). Then
Proof. For , by direct calculation, we have . In the following, we suppose that . From Lemma 6, we have For , we have From Lemmas 1 and 3, we have . Then from (49) we have . From the proof of Lemma 8, we have (). The result follows.
Theorem 22. Among all connected graphs on vertices with maximum clique size and (), the first four smallest spectral radii are obtained for , , , and (see Figures 1, 4, and 5), respectively.
Proof. Let be a connected graph with maximum clique size and vertices. Suppose that is a maximum clique of . From Lemmas 2, 4, and 20, we have
Thus, we only need to prove that if , , , . We distinguish the following three cases.
Case 1. There exists exactly one vertex outside of that is adjacent to one vertex of .
Subcase 1. Suppose that is a tree. If contains exactly one pendant vertex, then . If contains exactly two pendant vertices, then or . If contains three pendant vertices, then (see Figure 10). From Lemma 4, we have .
Subcase 2. Suppose that contains a cycle. If contains , then contains as a subgraph, where is obtained from by adding an edge between two pendant vertices. From Lemma 2, we have . If does not contain , then or contains as a proper subgraph. From Lemmas 2 and 7, we have . Note that . Thus, we have .
Case 2. There exists at least one vertex outside of that is adjacent to at least two vertices of . Then contains (see Figure 8) as a subgraph. From Lemmas 2, 7, and 21, we have .
Case 3. There exist at least two vertices outside of that are adjacent to some vertices of . Then contains or as a subgraph (see Figures 3 and 7). From Lemmas 2, 5, and 21, we have . Thus, from Lemma 2, we have .
4. Conclusion
In this paper, the first four graphs, which have the smallest values of the spectral radius among all connected graphs of order with maximum clique size , are determined.
Conflict of Interests
The authors declare that they have no conflict of 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 Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020).