The Smallest Spectral Radius of Graphs with a Given Clique Number

The first four smallest values of the spectral radius among all connected graphs with maximum clique size ω ≥ 2 are obtained.


Introduction
Let = ( ( ), ( )) be a simple connected graph with vertex set ( ) = {V 1 , V 2 , . . . , V } and edge set ( ). Its adjacency matrix ( ) = ( ) is defined as × matrix ( ), where = 1 if V is adjacent to V and = 0, otherwise. Denote by (V ) or (V ) the degree of the vertex V . 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 det( − ( )), which is denoted by Φ( ) or Φ( , ). Let be an eigenvector of corresponding to ( ). It will be convenient to associate with a labelling of in which vertex V is labelled (or V ). 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 I , be the set of all connected graphs of order with a maximum clique size , where 2 ≤ ≤ . It is easy to see that I , = { }. By direct calculation, we have ( ) = − 1. If ∈ I +1, , then, from the Perron-Frobenius theorem, the first − 1 smallest values of the spectral radius of I +1, are 1, ; (0 ≤ ≤ − 2), respectively, where 1, ; is the graph obtained from 1, by adding (0 ≤ ≤ − 2) 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 V be a vertex of the graph . Then the inequalities hold. If is connected, 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 V be a vertex of the graph and (V) the set of vertices adjacent to V.
Lemma 6 (see [12]). Let V be a vertex of , let (V) be the collection of circuits containing V, and let ( ) denote the set of vertices in the circuit . Then the characteristic polynomial Φ( ) satisfies where the first summation extends over those vertices adjacent to V, and the second summation extends over all ∈ (V).
Let be the tree on vertices obtained from −4 by attaching two new pendant edges to each end vertex of −4 , respectively.
Lemma 7 (see [13]). Suppose that ̸ = is a connected graph and V is an edge on an internal path of . Let V be the graph obtained from by subdivision of the edge V. Then ( V ) < ( ).

Main Results
Let 1 be the graph obtained from and a path 4 : V 1 V 2 V 3 V 4 by joining a vertex of and a nonpendant vertex, say, V 2 , of 4 by a path with length 2 and let 2 be the graph obtained from by attaching two pendant edges at two different vertices of (see Figure 3). Lemma 8. Let 1 and 2 be the graphs defined as above (see Figure 3). If ≥ 3, then ( 2 ) > ( 1 ).

Let
− , be the graph obtained from the kite graph − −1, (see Figure 1) and an isolated vertex V by adding an edge V V ( + 1 ≤ ≤ − 1) (see Figure 4). It is easy to see that +2 5, Figure 5).
Thus, ( Let 3 be the graph obtained from by attaching two pendant edges at some vertex of ; let 4 be the graph obtained from and 2 by adding two edges between two vertices of and two end vertices of 2 (see Figure 7).

Theorem 12.
Among all connected graphs on vertices with maximum clique size = 3 and ≥ 9, the first four smallest spectral radii are exactly obtained for −3,3 , Proof. Let be a connected graph with maximum clique size = 3 and ≥ 9 vertices. From Lemmas 2 and 7, we have Thus, we only need to prove that ( ) > ( Case 2. Suppose that there exists a vertex, say, , which does not belong to 3 , such that is adjacent to at least two vertices of 3 . Then contains * 4 as a proper subgraph, where * 4 is obtained from 4 by adding an edge between two disjoint vertices. From Lemmas 2 and 7, we have The Scientific World Journal 5 Case 3. Suppose that there uniquely exists a vertex which does not belong to 3 such that is adjacent to a vertex of 3 . We distinguish the following two cases.  is the graph obtained from 3 and a cycle by joining a vertex of 3 and a vertex of with a path and | ( )| ≤ (see Figure 7). Suppose that is labelled     Let 2 ( ≥ 4) be the graph as shown in Figure 8. Proof. Let be a connected graph with maximum clique size ≥ 4 and ≥ + 5 vertices. Suppose that is a maximum clique of . From Lemmas 2, 4, and 13, we have Thus, we only need to prove that ( ) > ( We distinguish the following three cases. 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 ̸ = − , , Lemma 15. Let 3 and 4 be the graphs defined as above (see Figure 7). Then The Scientific World Journal Proof. Let = ( 1 , 2 , . . . , ) be the Perron vector of 3 , where corresponds to V . From = ( 3 ) , we have ( 3 ) 2 = 2 1 + ( − 1) , From above equations, we have Similarly, we have ( 4 ) which is the largest root of equation Then we have, for > − 1, Thus, we have ( 3 ) < ( 4 ).
Case 2. Suppose that the two vertices outside of that are all adjacent to some vertices of . Note that ̸ = 2 , 3 , 4 . Then contains one of graphs 3 and 2 as a subgraph, where 3 is obtained from 3 by adding an edge between two pendant vertices. From Lemma 5, we have ( ) ≥ ( 3 ) > ( 4 ). From Lemmas 2 and 7, ( ) > ( 2 ) > ( 4 ). Let 5 be the graph obtained from 2 and an isolated vertex by adding an edge between a pendant vertex of 2 and the isolated vertex; let +1 3, and 6 be the graphs as shown in Figure 9.
Lemma 17. Let +1 3, and 5 be the graphs defined as above (see Figure 9). Then Proof. Let = ( 1 , 2 , . . . , ) be the Perron vector of +1 3, , where corresponds to V . It is easy to see that 1 = From above equations, we have Then for ≥ 4, we have The result follows.
Lemma 18. Let 5 and 6 be the graphs defined as above (see Figure 9). Then The Scientific World Journal 7 Proof. For = 4, by direct calculation, we have ( 6 ) > ( 5 ). In the following, we suppose that ≥ 5. Then, from Lemmas 2 and 3, we have > ( 5 ) > ( ) = − 1 ≥ 4. Let = ( 1 , 2 , . . . , ) be the Perron vector of 5 , where corresponds to V . From = ( 5 ) , we have From above equations, we have for > ( 5 ) > − 1 ≥ 4, Then, from Lemma 5, we have Let 7 be the graph obtained from 3 and an isolated vertex by adding an edge between V and the isolated vertex; let 8 be the graph obtained from 3 and an isolated vertex by adding an edge between V 2 and the isolated vertex; let 9 be the graph obtained from 3 and an isolated vertex by adding an edge between one pendant vertex and the isolated vertex; and let 10 be the graph obtained from +1 3, and an isolated vertex by adding an edge between V +1 and the isolated vertex (see Figure 10).
3, , and 5 be the graphs defined as above (see Figures 1,4,5,and 9). Among all connected graphs on vertices with maximum clique size and = + 3 ( ≥ 4), the first four smallest spectral radii are obtained for 3, , Thus, we only need to prove that ( ) > ( Proof. From Lemma 6, we have Then, we have

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 ≥ 2, are determined.