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

1. Introduction

Let G=(V(G),E(G)) be a simple connected graph with vertex set V(G)={v1,v2,…,vn} and edge set E(G). Its adjacency matrix A(G)=(aij) is defined as n×n matrix (aij), where aij=1 if vi is adjacent to vj and aij=0, otherwise. Denote by d(vi) or dG(vi) the degree of the vertex vi. It is well known that A(G) is a real symmetric matrix. Hence, the eigenvalues of A(G) can be ordered as
(1)λ1(G)≥λ2(G)≥⋯≥λn(G),
respectively. The largest eigenvalue of A(G) is called the spectral radius of G, denoted by ρ(G). It is easy to see that if G is connected, then A(G) is nonnegative irreducible matrix. By the Perron-Frobenius theory, ρ(G) has multiplicity one and exists a unique positive unit eigenvector corresponding to ρ(G). We refer to such an eigenvector corresponding to ρ(G) as the Perron vector of G.

Denote by Pn and Cn the path and the cycle 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). Let X be an eigenvector of G corresponding to ρ(G). It will be convenient to associate with X a labelling of G in which vertex vi is labelled xi (or xvi). 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 ω are given. And, in [5], it is shown that among all connected graphs with maximum clique size ω the minimum value of the spectral radius is attained for a kite graph PKn-ω,ω, where PKn-ω,ω is a graph on n vertices obtained from the path Pn-ω and the complete graph Kω by adding an edge between an end vertex of Pn-ω and a vertex of Kω (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 ω.

Kite graph PKn-ω,ω.

Let In,ω be the set of all connected graphs of order n with a maximum clique size ω, where 2≤ω≤n. It is easy to see that Iω,ω={Kω}. By direct calculation, we have ρ(Kω)=ω-1. If G∈Iω+1,ω, then, from the Perron-Frobenius theorem, the first ω-1 smallest values of the spectral radius of Iω+1,ω are PK1,ω;i (0≤i≤ω-2), respectively, where PK1,ω;i is the graph obtained from PK1,ω by adding i (0≤i≤ω-2) edges. So in the following, we consider that n≥ω+2.

2. Preliminaries

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

Lemma 1 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

Let v be a vertex of the graph G. Then the inequalities
(2)λ1(G)≥λ1(G-v)≥λ2(G)≥λ2(G-v)≥⋯≥λn-1(G-v)≥λn(G)
hold. If G is connected, then λ1(G)>λ1(G-v).

For the spectral radius of a graph, by the well-known Perron-Frobenius theory, we have the following.

Lemma 2.

Let G be a connected graph and H a proper subgraph of G. Then ρ(H)<ρ(G).

Lemma 3 (see [<xref ref-type="bibr" rid="B6">6</xref>, <xref ref-type="bibr" rid="B7">7</xref>]).

Let G be a graph on n vertices, then
(3)ρ(G)≤max{d(v):v∈V(G)}.
The equality holds if and only if G is a regular graph.

Let v be a vertex of a graph G and suppose that two new paths P=v(vk+1)vk⋯v2v1 and Q=v(ul+1)ul⋯u2u1 of lengths k and l (k≥l≥1) are attached to G at v(=vk+1=ul+1), respectively, to form a new graph Gk,l (shown in Figure 2), where v1,v2,…,vk and u1,u2,…,ul are distinct. Let
(4)Gk+1,l-1=Gk,l-u1u2+v1u1.
We call that Gk+1,l-1 is obtained from Gk,l by grafting an edge (see Figure 2).

Grafting an edge.

Lemma 4 (see [<xref ref-type="bibr" rid="B8">8</xref>, <xref ref-type="bibr" rid="B9">9</xref>]).

Let G be a connected graph on n≥2 vertices and v is a vertex of G. Let Gk,l and Gk+1,l-1 (k≥l≥1) be the graphs as defined above. Then ρ(Gk,l)>ρ(Gk+1,l-1).

Let v be a vertex of the graph G and N(v) the set of vertices adjacent to v.

Lemma 5 (see [<xref ref-type="bibr" rid="B10">10</xref>, <xref ref-type="bibr" rid="B11">11</xref>]).

Let G be a connected graph, and let u,v be two vertices of G. Suppose that v1,v2,…,vs∈N(v)∖(N(u)⋃{u}) (1≤s≤d(v)) and x=(x1,x2,…,xn) is the Perron vector of G, where xi corresponds to the vertex vi (1≤i≤n). Let G* be the graph obtained from G by deleting the edges vvi and adding the edges uvi (1≤i≤s). If xu≥xv, then ρ(G)<ρ(G*).

Lemma 6 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Let v be a vertex of G, 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
(5)Φ(G)=xΦ(G-v)-∑wΦ(G-v-w)-2∑Z∈φ(v)Φ(G-V(Z)),
where the first summation extends over those vertices w adjacent to v, and the second summation extends over all Z∈φ(v).

An internal path of a graph G is a sequence of vertices v1,v2,…,vk with k≥2 such that

the vertices in the sequence are distinct (except possibly v1=vk);

vi is adjacent to vi+1, (i=1,2,…,k-1);

the vertex degrees d(vi) satisfy d(v1)≥3,d(v2)=⋯=d(vk-1)=2 (unless k=2) and d(vk)≥3.

Let Wn be the tree on n vertices obtained from Pn-4 by attaching two new pendant edges to each end vertex of Pn-4, respectively.

Lemma 7 (see [<xref ref-type="bibr" rid="B13">13</xref>]).

Suppose that G≠Wn is a connected graph and uv is an edge on an internal path of G. Let Guv be the graph obtained from G by subdivision of the edge uv. Then ρ(Guv)<ρ(G).

3. Main Results

Let H1 be the graph obtained from Kω and a path P4:v1v2v3v4 by joining a vertex of Kω and a nonpendant vertex, say, v2, of P4 by a path with length 2 and let H2 be the graph obtained from Kω by attaching two pendant edges at two different vertices of Kω (see Figure 3).

H1 and H2.

Lemma 8.

Let H1 and H2 be the graphs defined as above (see Figure 3). If ω≥3, then ρ(H2)>ρ(H1).

Proof.

For 5≥ω≥3, by direct computations, we have ρ(H2)>ρ(H1). In the following, we suppose that ω≥6. From Lemma 6, we have
(6)Φ(H1)=(x+1)ω-2[x7-(ω-2)x6-(ω+4)x5hhhhhhhhhhh+(5ω-10)x4+(4ω+1)x3hhhhhhhhhhh-(5ω-10)x2-(2ω-1)x+ω-2]=(x-ω+2)ω-2g1(x).Φ(H2)=(x+1)ω-3[x5-(ω-3)x4-(2ω-1)x3hhhhhhhhhhh+(ω-5)x2+(2ω-3)x-ω+3]=(x+1)ω-3g2(x).

By direct calculation, we have
(7)g1(ω-1+1ω2)=-ω3+2ω2+6ω+13ω+26ω2-54ω3+26ω4+34ω5-54ω6+20ω7+20ω8-25ω9+5ω10+6ω11-5ω12+1ω14-20<0;g1(ω-1+2ω2)=ω4-6ω3+7ω2+26ω+66ω+166ω2-416ω3+224ω4+432ω5-832ω6+320ω7+560ω8-800ω9+160ω10+384ω11-320ω12+128ω14-91>0;g2(ω-1+2ω2)=-2ω+12ω-18ω2-8ω3+48ω4-48ω5-8ω6+64ω7-32ω8+32ω10<0.
From Lemmas 1 and 3, we have ω>ρ(H1)≥ρ(Kω)=ω-1≥λ2(H1) and ω>ρ(H2)≥ρ(Kω)=ω-1. Then from (7) we have ρ(H2)>ω-1+(2/ω2)>ρ(H1).

Let PKn-ω,ωi be the graph obtained from the kite graph PKn-ω-1,ω (see Figure 1) and an isolated vertex vn by adding an edge vnvi (ω+1≤i≤n-1) (see Figure 4). It is easy to see that PK5,ωω+2=H1 and PKn-ω,ωn-1=PKn-ω,ω.

Graph PKn-ω,ωi, where i=ω+1,…,n-1.

Let PK¯n-ω,ωn-2=PKn-ω,ωn-2+vn-1vn (see Figure 5).

Graph PK¯n-ω,ωn-2.

Lemma 9.

Let PKn-ω,ωi be the graphs defined as above (see Figure 4). Then
(8)ρ(Pn)<ρ(PKn-2,2n-2)<ρ(Cn)=ρ(Wn)<ρ(PKn-2,2n-3),hhhh(n≥10).

Proof.

Clearly, Pn=P2n-2,0, PKn-2,22=P2n-3,1. From Lemma 4, we have
(9)ρ(Pn)<ρ(PKn-2,2n-2)<ρ(Wn)=2=ρ(Cn).
For n≥10, from Lemma 2, we have ρ(PKn-2,2n-3)≥ρ(PK8,27)≈2.00659>ρ(Cn).

Let G1=PKn-3,3n-3-vn-1vn-2+vn-3vn-1, let G2=PKn-3,3n-3+vn-1vn, and let Cn-1,1 be the graph obtained from Cn-1 and an isolated vertex by adding an edge between some vertex of Cn-1 and the isolated vertex (see Figure 6).

Graphs G1, G2, Cn-1,1.

Theorem 10.

Among all connected graphs on n vertices with maximum clique size ω=2 and n≥10, the first four smallest spectral radii are exactly obtained for Pn, PKn-2,2n-2, Cn, Wn, and PKn-2,2n-3, respectively.

Proof.

Let G be a connected graph with maximum clique size ω=2 and n≥10 vertices. From Lemma 9, we have ρ(Pn)<ρ(PKn-2,2n-2)<ρ(Wn)=ρ(Cn)<ρ(PKn-2,2n-3). Thus, we only need to prove that ρ(G)>ρ(PKn-2,2n-3) if G≠Pn, PKn-2,2n-2, Wn, Cn, PKn-2,2n-3. If G is a tree, note that G≠Pn, PKn-2,2n-2, Wn, PKn-2,2n-3, then, from Lemma 4, we have ρ(G)>ρ(PKn-2,2n-3). If G contains some cycle as a subgraph, then, from Lemmas 2 and 7, we have ρ(G)≥ρ(Cn-1,1)>ρ(PKn-2,2n-3).

Lemma 11.

Let PKn-ω,ωi, PK¯n-ω,ωn-2, G1 and G2 be the graphs defined as above (see Figures 4, 5, and 6). Then
(10)ρ(PKn-3,3n-4)<min{ρ(PK¯n-3,3n-2),ρ(G1),ρ(G2)},hhhhhhhhhhhhhhhhhhhhhh(n≥8).

Proof.

For 8≤n≤11, by direct calculation, we have ρ(PKn-3,3n-4)<ρ(G1). If n≥12, from Lemmas 2 and 7, we have 2.23601<ρ(PK8,3)<ρ(PKn-3,3n-4)<ρ(PK9,38)<2.23808. From Lemma 6, we have
(11)Φ(PKn-3,3n-4)=(x5-4x3+3x)Φ(PKn-8,3)-(x4-2x2)Φ(PKn-8,3-vn-5)=f1(x)Φ(PKn-8,3)-f2(x)Φ(PKn-8,3-vn-5),Φ(G1)=(x5-4x3)Φ(PKn-8,3)-(x4-3x2)Φ(PKn-8,3-vn-5)=f3(x)Φ(PKn-8,3)-f4(x)Φ(PKn-8,3-vn-5).
Then we have
(12)f3(x)Φ(PKn-3,3n-4)-f1(x)Φ(G1)=(f1(x)f4(x)-f2(x)f3(x))Φ(PKn-8,3-vn-5)=(-x7+7x5-9x3)Φ(PKn-8,3-vn-5)=R1(x)Φ(PKn-8,3-vn-5).
For 2.23601<x<2.23808, we have
(13)f1(x)>2.236015-4×2.238083+3×2.23601≈17>0;f3(x)>2.236015-4×2.238083≈11>0;R1(x)>-2.238087+7×2.236015-9×2.238083≈9>0.
Note that from Lemma 2, ρ(PKn-8,3-vn-5)<ρ(PKn-3,3n-4) and 2.23601<ρ(PKn-3,3n-4)<2.23808. Then, we have
(14)f3(x)Φ(PKn-3,3n-4)>f1(x)Φ(G1),hhhhix∈[ρ(PKn-3,3n-4),2.23808).
Thus, ρ(PKn-3,3n-4)<ρ(G1). By similar method, we have for n≥8(15)ρ(PKn-3,3n-4)<ρ(PK¯n-3,3n-2),ρ(PKn-3,3n-4)<ρ(G2).

Let H3 be the graph obtained from Kω by attaching two pendant edges at some vertex of Kω; let H4 be the graph obtained from Kω and P2 by adding two edges between two vertices of Kω and two end vertices of P2 (see Figure 7).

Graphs H3, H4, and Fg.

Theorem 12.

Among all connected graphs on n vertices with maximum clique size ω=3 and n≥9, the first four smallest spectral radii are exactly obtained for PKn-3,3, PKn-3,3n-2, PKn-3,3n-3, PKn-3,3n-4, respectively.

Proof.

Let G be a connected graph with maximum clique size ω=3 and n≥9 vertices. From Lemmas 2 and 7, we have
(16)ρ(PKn-3,3n-4)>ρ(PKn-3,3n-3)>ρ(PKn-3,3n-2)>ρ(PKn-3,3).
Thus, we only need to prove that ρ(G)>ρ(PKn-3,3n-4) if G≠PKn-3,3, PKn-3,3n-2, PKn-3,3n-3, PKn-3,3n-4.

We distinguish the following three cases.

Case 1. If there exist at least two vertices outside of K3 that are adjacent to some vertices of K3, then we have that G contains either H2 (ω=3) or H3 (ω=3) as a proper subgraph. If G contains H2 (ω=3) as a proper subgraph, from Lemmas 2 and 7, we have
(17)ρ(G)>ρ(H2)≈2.30278>ρ(PK6,35)≈2.26542>ρ(PKn-3,3n-4),(ω=3).
If G contains H3 (ω=3) as a proper subgraph, from Lemmas 2 and 7, we have
(18)ρ(G)>ρ(H3)≈2.34292>ρ(PK6,35)>ρ(PKn-3,3n-4),(ω=3).

Case 2. Suppose that there exists a vertex, say, u, which does not belong to K3, such that u is adjacent to at least two vertices of K3. Then G contains C4* as a proper subgraph, where C4* is obtained from C4 by adding an edge between two disjoint vertices. From Lemmas 2 and 7, we have
(19)ρ(G)>ρ(C4*)≈2.56155>ρ(PK6,35)>ρ(PKn-3,3n-4).

Case 3. Suppose that there uniquely exists a vertex u which does not belong to K3 such that u is adjacent to a vertex of K3. We distinguish the following two cases.

Subcase 1. Suppose that G-V(K3) is a tree. If there exist two vertices u,r∈V(G-V(K3)) such that d(u)≥3 and d(r)≥3, then, from Lemmas 2, 4, and 7, we have ρ(G)>ρ(PKn-3,3n-4). If there exists only one vertex u∈V(G-V(K3)) such that d(u)≥4, then, from Lemmas 2, 7, and 11, we have ρ(G)≥ρ(G1)>ρ(PKn-3,3n-4). If there exists exactly one vertex u∈V(G-V(K3)) such that d(u)=3, note that G≠PKn-3,3n-2, PKn-3,3n-3, PKn-3,3n-4, then from Lemmas 2 and 7 we have ρ(G)>ρ(PKn-3,3n-4).

Subcase 2. Suppose that G-V(K3) contains cycle Cg as a subgraph. If g=3,4, then, from Lemmas 2, 7 and 11, we have ρ(G)≥ρ(PK¯n-3,3n-2)>ρ(PKn-3,3n-4) or ρ(G)≥ρ(G2)>ρ(PKn-3,3n-4). If g≥5, then, from Lemma 2, we can construct a graph Fg from G by deleting vertices such that ρ(G)≥ρ(Fg), where Fg is the graph obtained from K3 and a cycle Cg by joining a vertex of K3 and a vertex of Cg with a path and |V(Fg)|≤n (see Figure 7). Suppose that Cg is labelled v1,v2,…,vg satisfying vivi+1∈E(Cg), (1≤i≤g-1), v1vg∈E(Cg), and d(v1)=3. Then, from Lemmas 2 and 7, we have ρ(Fg-v2v3)>ρ(PKn-3,3n-4). Thus, we have ρ(G)>ρ(PKn-3,3n-4).

Lemma 13.

Let PKn-ω,ωi and PK¯n-ω,ωn-2 be the graphs defined as above (see Figures 4 and 5). Then ρ(PKn-ω,ωn-3)>ρ(PK¯n-ω,ωn-2) (ω≥4).

Proof.

Let X=(x1,x2,…,xn)T be the Perron vector of PK¯n-ω,ωn-2, where xi corresponds to vi. It is easy to prove that xn=xn-1. From AX=ρ(PK¯n-ω,ωn-2)X, we have
(20)xn-2=(ρ(PK¯n-ω,ωn-2)-1)xn,xn-3=[ρ(PK¯n-ω,ωn-2)(ρ(PK¯n-ω,ωn-2)-1)-2]xn.
From Lemma 2, for ω≥4 we have ρ(PK¯n-ω,ωn-2)≥ρ(Kω)=ω-1≥3. Then
(21)ρ(PKn-ω,ωn-3)-ρ(PK¯n-ω,ωn-2)≥XTA(PKn-ω,ωn-3)X-XTA(PK¯n-ω,ωn-2)X=2xn(xn-3-xn-2-xn)=2[ρ(PK¯n-ω,ωn-2)(ρ(PK¯n-ω,ωn-2)-2)-2]xn≥2xn>0.
So, ρ(PKn-ω,ωn-3)>ρ(PK¯n-ω,ωn-2).

Let Mω2 (ω≥4) be the graph as shown in Figure 8.

Graph Mω2.

Theorem 14.

Among all connected graphs on n vertices with maximum clique size ω≥4 and n≥ω+5, the first four smallest spectral radii are exactly obtained for PKn-ω,ω, PKn-ω,ωn-2, PK¯n-ω,ωn-2, PKn-ω,ωn-3, respectively.

Proof.

Let G be a connected graph with maximum clique size ω≥4 and n≥ω+5 vertices. Suppose that Kω is a maximum clique of G. From Lemmas 2, 4, and 13, we have
(22)ρ(PKn-ω,ωn-3)>ρ(PK¯n-ω,ωn-2)>ρ(PKn-ω,ωn-2)>ρ(PKn-ω,ω).
Thus, we only need to prove that ρ(G)>ρ(PKn-ω,ωn-3) if G≠PKn-ω,ω, PKn-ω,ωn-2, PK¯n-ω,ωn-2, PKn-ω,ωn-3. We distinguish the following three cases.

Case 1. If there exist at least two vertices outside of Kω that are adjacent to some vertices of Kω, then G contains either H2 or H3 as a proper subgraph. If G contains H2 as a proper subgraph, from Lemmas 2, 7, and 8, we have
(23)ρ(G)>ρ(H2)>ρ(H1)≥ρ(PKn-ω,ωn-3).
If G contains H3 as a proper subgraph, from Lemmas 2, 5, 7, and 8, we have
(24)ρ(G)>ρ(H3)>ρ(H2)>ρ(H1)≥ρ(PKn-ω,ωn-3).

Case 2. Suppose that there exists a vertex, say, u, which does not belong to Kω, such that u is adjacent to at least two vertices of Kω. From Lemmas 2, 7, and 8, we have
(25)ρ(G)>ρ(Mω2)>ρ(H4)>ρ(H2)>ρ(H1)≥ρ(PKn-ω,ωn-3).

Case 3. Suppose that there uniquely exists a vertex u which does not belong to Kω such that u is adjacent to a vertex of Kω. If G-V(Kω) is a tree, note that G≠PKn-ω,ω, PKn-ω,ωn-2, PKn-ω,ωn-3, then, from Lemmas 2, 4, and 7, we have ρ(G)>ρ(PKn-ω,ωn-3). Suppose that G-V(Kω) contains cycle Cg as a subgraph. If g=3, note that G≠PK¯n-ω,ωn-2, then, from Lemmas 2 and 7, we have ρ(G)>ρ(G*)>ρ(PKn-ω,ωn-3), where G*=PKn-ω,ωn-3+vn-1vn. If g≥4, then by the similar reasoning as that of Subcase 2 of Case 3 of Theorem 12, we have ρ(G)>ρ(PKn-ω,ωn-3).

Lemma 15.

Let H3 and H4 be the graphs defined as above (see Figure 7). Then
(26)ρ(H4)>ρ(H3)(ω≥3).

Proof.

Let X=(x1,x2,…,xn)T be the Perron vector of H3, where xi corresponds to vi. From AX=ρ(H3)X, we have
(27)ρ(H3)x1=x2,ρ(H3)x2=2x1+(ω-1)xω,ρ(H3)xω=(ω-2)xω+x2.
From above equations, we have
(28)ρ3(H3)-(ω-2)ρ2(H3)-(ω+1)ρ(H3)+2ω-4=0.
Let
(29)r1(x)=x3-(ω-2)x2-(ω+1)x+2ω-4.
Then
(30)r1(ω-1)=-2<0.
For x>ω-1 and ω≥3, we have
(31)r1′(x)=3x2-2(ω-2)x-(ω+1)>0.
Note that ρ(H3)>ρ(Kω)=ω-1. From (30) and (31), we have ρ(H3) which is the largest root of equation r1(x)=0. Similarly, we have ρ(H4) which is the largest root of equation
(32)r2(x)=x3-(ω-1)x2-2x+2ω-4=0.
Then we have, for x>ω-1,
(33)r1(x)-r2(x)=x2-(ω-1)x>0.
Thus, we have ρ(H3)<ρ(H4).

Theorem 16.

Let G be a graph on n vertices with maximum clique size ω≥3 and n=ω+2. Let PK2,ω, H2, H3, and H4 be the graphs defined as above (see Figures 1, 3 and 7). The first four smallest spectral radii are obtained for PK2,ω, H2, H3, H4, respectively.

Proof.

From Lemmas 2, 5, 8, and 15, we have
(34)ρ(H4)>ρ(H3)>ρ(H2)>ρ(H1)>ρ(PK2,ω).
Thus, we only need to prove that, for G≠PK2,ω, H2, H3, and H4, ρ(G)>ρ(H4). We distinguish the following two cases.

Case 1. Suppose that there exists exactly one vertex outside of Kω that is adjacent to at least two vertices of Kω. Then G contains Mω2 (see Figure 8) as a subgraph. From Lemmas 2 and 7, we have ρ(Mω2)>ρ(H4).

Case 2. Suppose that the two vertices outside of Kω that are all adjacent to some vertices of Kω. Note that G≠H2, H3, H4. Then G contains one of graphs H¯3 and Mω2 as a subgraph, where H¯3 is obtained from H3 by adding an edge between two pendant vertices. From Lemma 5, we have ρ(G)≥ρ(H¯3)>ρ(H4). From Lemmas 2 and 7, ρ(G)>ρ(Mω2)>ρ(H4).

Let H5 be the graph obtained from H2 and an isolated vertex by adding an edge between a pendant vertex of H2 and the isolated vertex; let PK¯3,ωω+1 and H6 be the graphs as shown in Figure 9.

Graphs PK¯3,ωω+1, H5, and H6.

Lemma 17.

Let PK¯3,ωω+1 and H5 be the graphs defined as above (see Figure 9). Then
(35)ρ(H5)>ρ(PK¯3,ωω+1),(ω≥4).

Proof.

Let X=(x1,x2,…,xn)T be the Perron vector of PK¯3,ωω+1, where xi corresponds to vi. It is easy to see that x1=x5. From AX=ρ(PK¯3,ωω+1)X, we have
(36)ρ(PK¯3,ωω+1)x1=x1+x2,ρ(PK¯3,ωω+1)x2=2x1+x3,ρ(PK¯3,ωω+1)x3=x2+(ω-1)x4,ρ(PK¯3,ωω+1)x4=x3+(ω-2)x4.
From above equations, we have
(37)x2=(ρ(PK¯3,ωω+1)-1)x1,x4=ρ2(PK¯3,ωω+1)-ρ(PK¯3,ωω+1)-2ρ(PK¯3,ωω+1)-ω+2x1.
Then for ω≥4, we have
(38)ρ(H5)-ρ(PK¯3,ωω+1)≥XTA(H5)X-XTA(PK¯3,ωω+1)X=2x1(x4-x2-x1)=2(ω-3)ρ(PK¯3,ωω+1)-2ρ(PK¯3,ωω+1)-ω+2x1>0.
The result follows.

Lemma 18.

Let H5 and H6 be the graphs defined as above (see Figure 9). Then
(39)ρ(H6)>ρ(H5),(ω≥4).

Proof.

For ω=4, by direct calculation, we have ρ(H6)>ρ(H5). In the following, we suppose that ω≥5. Then, from Lemmas 2 and 3, we have ω>ρ(H5)>ρ(Kω)=ω-1≥4. Let X=(x1,x2,…,xn)T be the Perron vector of H5, where xi corresponds to vi. From AX=ρ(H5)X, we have
(40)ρ(H5)x1=x2,ρ(H5)x2=x1+x3,ρ(H5)x3=x2+x4+(ω-2)x6,ρ(H5)x4=x3+x5+(ω-2)x6,ρ(H5)x5=x4,ρ(H5)x6=x3+x4+(ω-3)x6.
From above equations, we have for ω>ρ(H5)>ω-1≥4,
(41)x6=ρ2(H5)-1ρ(H5)-ω+3x1+(ρ2(H5)+ρ(H5))(ρ2(H5)-1)-ρ2(H5)(ρ(H5)-ω+3)(ρ2(H5)+ρ(H5)-1)x1>ρ2(H5)-13x1>ρ(H5)x1=x2.
Then, from Lemma 5, we have ρ(H6)=ρ(H5-v1v2+v1v6)>ρ(H5).

Let H7 be the graph obtained from H3 and an isolated vertex by adding an edge between vω and the isolated vertex; let H8 be the graph obtained from H3 and an isolated vertex by adding an edge between v2 and the isolated vertex; let H9 be the graph obtained from H3 and an isolated vertex by adding an edge between one pendant vertex and the isolated vertex; and let H10 be the graph obtained from PK3,ωω+1 and an isolated vertex by adding an edge between vω+1 and the isolated vertex (see Figure 10).

Graphs H7, H8, H9, H10.

Theorem 19.

Let PK3,ω, PK3,ωω+1, PK¯3,ωω+1, and H5 be the graphs defined as above (see Figures 1, 4, 5, and 9). Among all connected graphs on n vertices with maximum clique size ω and n=ω+3 (ω≥4), the first four smallest spectral radii are obtained for PK3,ω, PK3,ωω+1, PK¯3,ωω+1, and H5, respectively.

Proof.

From Lemmas 2, 4, and 17, we have
(42)ρ(H5)>ρ(PK¯3,ωω+1)>ρ(PK3,ωω+1)>ρ(PK3,ω).
Thus, we only need to prove that ρ(G)>ρ(H5) if G≠PK3,ω, PK3,ωω+1, PK¯3,ωω+1, and H5. We distinguish the following four cases.

Case 1. There exists exactly one vertex outside of Kω that is adjacent to only one vertex of Kω. Then G must be one of graphs PK3,ω, PK3,ωω+1, and PK¯3,ωω+1.

Case 2. There exists one vertex outside of Kω that is adjacent to at least two vertices of Kω. Then G contains Mω2 (see Figure 8) as a proper subgraph. From Lemmas 2 and 7, we have ρ(G)>ρ(Mω2)>ρ(H5).

Case 3. If there exactly exist two vertices outside of Kω that are adjacent to some vertices of Kω, then G contains H5 or H9 (see Figures 9 and 10) as a subgraph. If G contains H9 as a subgraph, then, from Lemmas 2 and 5, we have ρ(G)≥ρ(H9)>ρ(H5). If G contains H5 as a subgraph, note that G≠H5, then, from Lemma 2, we have ρ(G)>ρ(H5).

Case 4. If there exist three vertices outside of Kω that are adjacent to some vertices of Kω, then G contains one of graphs H6, H7, and H8 (see Figures 9 and 10) as a subgraph. From Lemmas 5 and 18, we have ρ(H8)>ρ(H7)>ρ(H6)>ρ(H5). Then, from Lemma 2, we have ρ(G)>ρ(H5).

Lemma 20.

Let PKn-ω,ωi and PK¯4,ωω+2 be the graphs defined as above (see Figures 4 and 5). Then
(43)ρ(PK4,ωω+1)>ρ(PK¯4,ωω+2),(ω≥4).

Proof.

From Lemma 6, we have
(44)Φ(PK¯4,ωω+2)=(x4-4x2-2x+1)Φ(Kω)-(x3-3x-2)Φ(Kω-1)=f5(x)Φ(Kω)-f6(x)Φ(Kω-1);Φ(PK4,ωω+1)=(x4-3x2+1)Φ(Kω)-(x3-x)Φ(Kω-1)=f7(x)Φ(Kω)-f8(x)Φ(Kω-1).
Then, we have
(45)f7(x)Φ(PK¯4,ωω+2)-f5(x)Φ(PK4,ωω+1)=(f5(x)f8(x)-f6(x)f7(x))Φ(Kω-1)=(x5-5x3-4x2+2x+2)Φ(Kω-1)=R2(x)Φ(Kω-1).
For x>ω-1 (ω≥4), we have
(46)f5(x)>0,f7(x)>0,R2(x)>0,Φ(Kω-1)>0.
From Lemma 2, we have ρ(PK¯4,ωω+2)>ρ(Kω)=ω-1 and ρ(PK4,ωω+1)>ρ(Kω)=ω-1. Thus, for x>ω-1 (ω≥4), we have f7(x)Φ(PK¯4,ωω+2)-f5(x)Φ(PK4,ωω+1)>0. Then ρ(PK4,ωω+1)>ρ(PK¯4,ωω+2), (ω≥4).

Lemma 21.

Let PKn-ω,ωi and H2 be the graphs defined as above (see Figures 3 and 4). Then
(47)ρ(H2)>ρ(PK4,ωω+1),(ω≥3).

Proof.

For ω=3,4,5, by direct calculation, we have ρ(H2)>ρ(PK4,ωω+1). In the following, we suppose that ω≥6. From Lemma 6, we have
(48)Φ(PK4,ωω+1)=(x+1)ω-2[x6-(ω-2)x5-(ω+3)x4hhhhhhhhhhh+(4ω-8)x3+(3ω-1)x2hhhhhhhhhhhi-(2ω-4)x-ω+1]=(x+1)ω-2g3(x).
For ω≥6, we have
(49)g3(ω-1+1ω2)=-ω2-13ω+4ω2+17ω3-24ω4+6ω5+14ω6-16ω7+2ω8+5ω9-4ω10+1ω12+7<0;g3(ω-1+2ω2)=ω3-5ω2+2ω-58ω+20ω2+108ω3-192ω4+48ω5+192ω6-256ω7+32ω8+160ω9-128ω10+64ω12+24>0.
From Lemmas 1 and 3, we have ω>ρ(PK4,ωω+1)≥ρ(Kω)=ω-1≥λ2(PK4,ωω+1). Then from (49) we have ω-1+2/ω2>ρ(PK4,ωω+1)>ω-1+1/ω2. From the proof of Lemma 8, we have ρ(H2)>ω-1+2/ω2 (ω≥6). The result follows.

Theorem 22.

Among all connected graphs on n vertices with maximum clique size ω and n=ω+4 (ω≥4), the first four smallest spectral radii are obtained for PK4,ω, PK4,ωω+2, PK¯4,ωω+2, and PK4,ωω+1 (see Figures 1, 4, and 5), respectively.

Proof.

Let G be a connected graph with maximum clique size ω≥4 and n=ω+4 vertices. Suppose that Kω is a maximum clique of G. From Lemmas 2, 4, and 20, we have
(50)ρ(PK4,ωω+1)>ρ(PK¯4,ωω+2)>ρ(PK4,ωω+2)>ρ(PK4,ω).
Thus, we only need to prove that ρ(G)>ρ(PK4,ωω+1) if G≠PK4,ω, PK4,ωω+2, PK¯4,ωω+2, PK4,ωω+1. We distinguish the following three cases.

Case 1. There exists exactly one vertex outside of Kω that is adjacent to one vertex of Kω.

Subcase 1. Suppose that G-V(Kω) is a tree. If G contains exactly one pendant vertex, then G=PK4,ω. If G contains exactly two pendant vertices, then G=PK4,ωω+1 or G=PK4,ωω+2. If G contains three pendant vertices, then G=H10 (see Figure 10). From Lemma 4, we have ρ(H10)>ρ(PK4,ωω+1).

Subcase 2. Suppose that G-V(Kω) contains a cycle. If G-V(Kω) contains C4, then G contains H11 as a subgraph, where H11 is obtained from PK4,ωω+1 by adding an edge between two pendant vertices. From Lemma 2, we have ρ(H11)>ρ(PK4,ωω+1). If G-V(Kω) does not contain C4, then G=PK¯4,ωω+2 or G contains PK¯3,ωω+1 as a proper subgraph. From Lemmas 2 and 7, we have ρ(PK¯3,ωω+1)>ρ(H11)>ρ(PK4,ωω+1). Note that G≠PK¯4,ωω+2. Thus, we have ρ(G)>ρ(PK4,ωω+1).

Case 2. There exists at least one vertex outside of Kω that is adjacent to at least two vertices of Kω. Then G contains Mω2 (see Figure 8) as a subgraph. From Lemmas 2, 7, and 21, we have ρ(G)>ρ(Mω2)>ρ(H2)>ρ(PK4,ωω+1).

Case 3. There exist at least two vertices outside of Kω that are adjacent to some vertices of Kω. Then G contains H2 or H3 as a subgraph (see Figures 3 and 7). From Lemmas 2, 5, and 21, we have ρ(H3)>ρ(H2)>ρ(PK4,ωω+1). Thus, from Lemma 2, we have ρ(G)>ρ(PK4,ωω+1).

4. Conclusion

In this paper, the first four graphs, which have the smallest values of the spectral radius among all connected graphs of order n with maximum clique size ω≥2, 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).

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