Graph theory is a branch of discrete mathematics, Its research object is abstracted from the actual problem. For example, the geometric structure of an electrical network can be represented as a corresponding line graph. In the graph, the properties of circuit elements are ignored, the length and bending of edges are not important, but the connection between nodes and branches is highlighted. Each element in the network is replaced by a line segment, which is called a branch, and the endpoint of each element or the point connected by several elements is represented by an origin, which is called a node. The set of points and lines is called a network graph and is represented by G. Let G = (V, E) be a simple connected graph with n vertices, m edges [1]. Let P_n, C_n and S_n be the path, the cycle and the star with n vertices, respectively [1]. Let N_G(v) = {u|uv ∈ E(G)}, denote by d_G(v) = |N_G(v)| the degree of the vertex v of G. We know that L(G) = D(G) − A(G) is the Laplacian matrix of G, and A(G) is (0, 1) adjacency matrix, D(G) is degree diagonal matrix. Corresponding to the Laplacian matrix, Q(G) = D(G) + A(G) is called the signless Laplacian matrix of a graph [2]. The Laplacian characteristic polynomials and signless Laplacian characteristic are defined as the following

For G, H, if c_i(G) ≤ c_i(H),i = 1, 2, …, n, we call that G ≼′ H. If φ_i(G) ≤ φ_i(H), i = 1, 2, …, n, we call that G ≼ H [3, 4].

Denote by g_n,m the set of simple connected graphs of order n and size m. If m = n − 1 + c, G denotes a c-cyclic graph. If c = 0, 1, 2, and 3, G represents a tree, unicyclic graph, bicyclic graph and tricyclic graph, respectively [1]. Recently, with further research on the power system network, the study of the structure and properties of the partial ordering sets (gn,m,≼′) and (g_n,m, ≼) have attracted much attention. For m = n − 1, Mohar [5] proved that there is unique maximal element and unique minimal element in (gn,n-1,≼′). Since L(G; λ) = Q(G; λ) for bipartite graph, then (g_{n,n − 1}, ≼) has the same structure and properties as (gn,n-1,≼′). For m = n, Stevanović and Ilić [6] showed that there is also unique maximal element and unique minimal element in (gn,n,≼′). But for (g_n,n, ≼), Li et al. [7] given the extremal elements in (g_n,n, ≼). He and Shan [8] obtained the unique minimal element in (gn,n+1,≼′), and in Zhang and Zhang [3], two minimal elements in (g_{n,n + 1}, ≼) were determined by Zhang and Zhang. For simplicity, denote the class of connected tricyclic graphs order n, i.e., g_{n,n + 2} by T_n [9]. Pai et al. [10] characterized the unique minimal element in (Tn,≼′). Based on these works, we focus on the structure and properties of the partial ordering sets (T_n, ≼).

In this section, we introduce some graphic transformations and lemmas, which will be used to prove our main results.

If a connected graph has only one cycle whose length is odd, the graph is odd unicyclic. If the components of a spanning subgraph of G are trees or odd unicyclic graphs, the subgraph is called a TU-subgraph of G [3]. Let H be a TU-subgraph of G, which contains c odd unicyclic graphs and s trees T₁, …, T_s of orders n₁, …, n_s, respectively. So the weight of H ω(H)=4c∏i=1sni. If there contains no tree in H, so ω(H) = 4^c. If H is empty graph,there is no H, so ω(H) = 0. We can express the signless Laplacian coefficients φ_i(G) by the weight of TU-subgraphs of G [11].

Lemma 2.1. [12] Let Q(G;λ)=det(λI-Q(G))=∑i=0n(-1)iφi(G)λn-i be the characteristic polynomial of the signless Laplacian matrix of a graph G of order n. Then φi(G)=∑Hiω(Hi), i = 1, …, n, where the summation runs over all TU-subgraph H_i of G with i edges.

Definition 1. [8] Let G be a simple connected graph with n vertices and uv be a non-pendent edge, which is not contained in any cycles of G. Let G_uv = G − {vx|x ∈ N_G(v)\{u}} + {ux|x ∈ N_G(v)\{u}}. We say that G_uv is an α-transformation of G.

Lemma 2.2. [3] Let G be a connected graph of order n ≥ 4, and G_uv be obtained from G by α-transformation. Then G_uv ≼ G, i.e., φ_i(G_uv) ≤ φ_i(G), i = 0, 1, …, n, with equality if and only if either i ∈ {0, 1, n} when G is non-bipartite, or i ∈ {0, 1, n − 1, n} for otherwise.

The proof of the following lemma can be found in many places in the literature (see, such as [13]).

Lemma 2.3. [14] L(G; λ) = Q(G; λ) if and only if the graph G is bipartite.

Lemma 2.4. [15] Let f(λ) and g(λ) be two real polynomials arranged according to decreasing exponents. If their coefficients are alternate about positive and negative, then the coefficients of f(λ)g(λ) also are alternate about positive and negative.

Let G be a connected graph with at least one cycle, the base of G is represented by G^, which is the minimal connected subgraph containing all the cycles of G [16]. So G^ is the unique subgraph of G, which contains no pendant vertex. G can be obtained from G^ by planting trees to some vertices of G^ [17]. Hoffman and Smith [18] define an internal path of G as a walk u₀u₁…u_s(s ≥ 1),and the vertices u₀, u₁, …, u_{s − 1} are distinct, d(u₀) > 2, d(u_s) > 2, and d(u_i) = 2, whenever 0 < i < s. An internal path is closed, if u₀ = u_s.

Definition 2. [19] Let G = (V, E) be a connected graph and the base of G is G^. Let u, v, w be three consecutive vertices in an internal path of length at least 4 of G^, which satisfy N_G(u) ∩ N_G(v) = ∅, N_G(w) ∩ N_G(v) = ∅ and N_G(u) ∩ N_G(w) = {v}. We can delete all edges vz for z ∈ N_G(v)\{u, w}, wz for z ∈ N_G(w) and add all edges uz for z ∈ (N_G(v) ∪ N_G(w))\{u, v} from G and get the graph G′(u, v, w). G to G′(u, v, w) is called a β-transformation of G.

Lemma 2.5. Let G = (V, E) be a connected graph and the base of G is G^. Let u, v, w be three consecutive vertices in an internal path of length at least 4 of G^, and G′(u, v, w) be a graph obtained from G by β-transformation [19]. So G′(u, v, w) ≼ G, that is, φi(G′(u,v,w))≤φi(G) for i ∈ {0, 1, 2, …, n}, with equality if and only if i ∈ {0, 1} when G is non-bipartite, and i ∈ {0, 1, n} when G is bipartite.

Proof: φ0(G′(u,v,w))=φ0(G)=1 and φ1(G′(u,v,w))=φ1(G)=2|E|. Moreover, φn(G′(u,v,w))=φn(G)=0 for bipartite graph. Now assume that 2 ≤ i ≤ n. Let H and H be the set of all TU-subgraphs of G′(u, v, w) and G with i edges, respectively. For an arbitrary TU-subgraph H′ ∈ H, denote by the R′ connected component of H′ containing u [3]. Let f : H → H with H′ → H = f(H′), where V(H) = V(H′) and

Then f is injective from H → H.

Case 1. u, v, w belongs the component S′. So f(S′) is a component of H, which in the same order as S′. Then ω(H) = ω(H′).

Case 2. u, v, w belong to at least two components of H′.

Case 2.1. u is not in an odd unicyclic component of H′. Then u is contained in a tree component of H′. Assume that there exist x₁ + 1 vertices in the connected component which contains u in H − uv [3], x₂ + 1 vertices in the connected component which contains w in H − wv and x₃ + 1 vertices in the connected component which contains v in H − uv − vw, where x₁, x₂, x₃ ≥ 0. Let N indicate the weight of the components of H′, which contain no u, v, w.

(i) If uv ∈ E(H′)anduw ∉ E(H′), then

(ii) If uv ∉ E(H′)anduw ∈ E(H′), then

(iii) If uv ∉ E(H′)anduw ∉ E(H′), then

Case 2.2. u is in an odd unicyclic component S′ of H′. Let C′ be a subgraph of S′, which corresponds to an odd cycle C in G.

(i) If uv ∉ E(H′), uw ∉ E(H′), and C = C′, let S be the component containing C in H. So there are the same components in H′ and H, except for S′, {v}, {w} in H′, which correspond to the component S containing u, two components S₁ containing v and S₂ containing w of order at least 1, respectively, in H. If uv ∉ E(H′), uw ∉ E(H′), and C ≠ C′. So there are the same components in H′ and H, except for S′, {v}, {w} in H′, which correspond to two tree components S₁ containing u, w of order at least 4 since u, v, w are three consecutive vertices in an internal path of length at least 4 of G^, and S₂ containing v of order at least 1, in H. So

(ii) If uv ∉ E(H′), uw ∈ E(H′) or uv ∈ E(H′), uw ∉ E(H′), and C = C′, So there are the same components in H′ and H, except for S′, {v} or {w} in H′, which correspond to an odd unicyclic component S containing C and a tree component S₁ containing v, w of order at least 2. So

If uv ∉ E(H′), uw ∈ E(H′) or uv ∈ E(H′), uw ∉ E(H′), and C ≠ C′, So there are the same components in H′ and H, except for S′, {v} or {w} in H′, which correspond to a tree component S containing u, v, w of order at least 5. So

Then by Lemma 2.1, we have φi(G′(u,v,w))=∑Hi′∈gω(Hi′)≤∑Hi∈gω(Hi)=φi(G)

Hence the results hold.     □

Similarly, we can prove the following result.

Lemma 2.6. [19] Let G = (V, E) be a connected graph with base G^. Let u, v, w be three consecutive vertices in an internal path P = u₁u₂…u_k with k = 4 of G^ and u1uk∉E(G^). Let G′(u, v, w) be a graph obtained from G by β-transformation, then G′(u, v, w) ≼ G, that is, φi(G′(u,v,w))≤φi(G) for i ∈ {0, 1, 2, …, n}, with equality if and only if i ∈ {0, 1} when G is non-bipartite, and i ∈ {0, 1, n} when G is bipartite.

By Li et al. [20], There are the following four types of bases in tricyclic graphs(as shown in Figures 1–4): Gj3(j=1,…,7),Gj4(j=1,…,4),Gj6(j=1,…,3) and G17. Let

Then Tn=Tn3∪Tn4∪Tn6∪Tn7.

Let T13(n-7,0,0,0,0,0,0),T14(n-6,0,0,0,0,0),T16(n-5,0,0,0,0) and T17(n-4,0,0,0) be the graphs as shown in Figure 5.

Lemma 2.7. [10]

For i = 3, 4, 6, 7, let Tni,e (resp., Tni,o) be the set of bipartite tricyclic graphs (resp., non-bipartite tricyclic graphs) in Tni, then Tni=Tni,e∪Tni,o. From lemmas 2.3 and 2.7, we get

Corollary 2.8. [10]

Theorem 2.9. [10] Let G be a connected tricyclic graph on n vertices and i be an integer, 0 ≤ i ≤ n. Then ci(G)≥ci(T17(n-4,0,0,0)).

Repeated by lemmas 2.2, 2.5, and 2.6, we get the following conclusion

Theorem 2.10. Let G be a graph in Tn3,o∪Tn4,o∪Tn6,o∪Tn7,o. So there is a tricyclic graph G′ with order n, such that G′ ≼ G,. The base of G′ is one of graphs in {Ti3|j=1,2,…,9}∪{Ti4|j=1,2,…,20}∪{Ti6|j=1,2,…,24}∪{Ti7|j=1,2,…,7}(these base graphs are as shown in Figures 7–9.

Now we consider the minimal element in the partial ordering set (T_n, ≼).

For i = 1, 2, …, 9, let Ti3(s1,s2,…,s|Ti3|) be the graph obtained from Ti3 (as shown in Figure 6) by attaching s_j pendent edges at uj(j=1,2,…,|Ti3|), where n=s1+s2+⋯+s|Ti3|+|Ti3|.

Lemma 3.1. For j = 1, 2, …, 9, Tj3(s1+s2+⋯+s|Tj3|,0,…,0)≼Tj3(s1,s2,…,s|Tj3|), that is, ϕi(Tj3(s1+s2+⋯+s|Tj3|,0,…,0))≤ϕi(Tj3(s1,s2,…,s|Tj3|)),i=0,1,…,n. The equality holds if and only if s2=⋯=s|Tj3|=0.

Proof: For convenience, let G=Tj3(s1,s2,…,s|Tj3|) and G′=Tj3(s1+s2+⋯+s|Tj3|,0,…,0) for j = 1, 2, …, 9. Note that ϕ0(G)=1=ϕ0(G′),ϕ1(G)=2(n+2)=ϕ1(G′). For 2 ≤ i ≤ n, let H and H be the set of all TU-subgraphs of G′ and G with exactly i edges, respectively [3]. Let

Similarly for H⁽¹⁾, H⁽²⁾, and H⁽³⁾. We only prove the case for j = 1, the others can be proved similarly.

Let f : H → H with H′ → H = f(H′), where V(H) = V(H′) and

for R′ being a component of H′ containing u₁. Obviously, f is injective and f(H′^(k)) ⊆ H^(k) for j = 1, 2, 3. From the procedure of proof in Theorem 3.1 [10], we have

Note that H′⁽³⁾ = ∅ for j = 1. For H′ ∈ H′⁽²⁾, without loss of generality, we assume that R′ contains C₃ = u₁u₂u₃u₁ as a subgraph. Let R be the component of H corresponding to R′, obviously, R also contains C₃ = u₁u₂u₃u₁. It is obvious that H′, H have the same number of components and the product of the order of components which contain no u_i(i = 1, 2, …, 7) of H′ is the same as H. The order of the tree components of H′, which include at least one of u_i(i = 4, …, 7) are no more than the corresponding ones of H, then ω(f(H′)) ≥ ω(H′). Hence

The equality holds if and only if s₂ = ⋯ = s₇ = 0.     □

For i = 1, 2, …, 20, let Ti4(s1,s2,…,s|Ti4|) be the graph obtained from Ti4 (as shown in Figure 7) by attaching s_j pendent edges at uj(j=1,2,…,|Ti4|), where n=s1+s2+⋯+s|Ti4|+|Ti4|. Similar to the proof of Lemma 3.1, we have

Lemma 3.2. For j = 1, 2, …, 20, Tj4(s1+s2+⋯+s|Tj4|,0,…,0)≼Tj4(s1,s2,…,s|Tj4|), that is, ϕi(Tj4(s1+s2+⋯+s|Tj4|,0,…,0))≤ϕi(Tj4(s1,s2,…,s|Tj4|)),i=0,1,…,n. The equality holds if and only if s2=⋯=s|Tj4|=0.

For i = 1, 2, …, 7, let Ti7(s1,s2,…,s|Ti7|) be the graph obtained from Ti7 (as shown in Figure 8) by attaching s_j pendent edges at uj(j=1,2,…,|Ti7|), where n=s1+s2+⋯+s|Ti7|+|Ti7|. Similar to the proof of Lemma 3.1, we have

Lemma 3.3. For j = 1, 2, …, 7, Tj7(s1+s2+⋯+s|Tj7|,0,…,0)≼Tj7(s1,s2,…,s|Tj7|), that is, ϕi(Tj7(s1+s2+⋯+s|Tj7|,0,…,0))≤ϕi(Tj7(s1,s2,…,s|Tj7|)),i=0,1,…,n. The equality holds if and only if s2=⋯=s|Tj7|=0.

For i = 1, 2, …, 24, let Ti6(s1,s2,…,s|Ti6|) be the graph obtained from Ti6 (as shown in Figure 9) by attaching s_j pendent edges at uj(j=1,2,…,|Ti6|), where n=s1+s2+⋯+s|Ti6|+|Ti6|. Similar to the proof of Lemma 3.1, we have

Lemma 3.4. For j = 1, 2, …, 24, Tj6(s1+s2+⋯+s|Tj6|,0,…,0)≼Tj6(s1,s2,…,s|Tj6|), that is, ϕi(Tj6(s1+s2+⋯+s|Tj6|,0,…,0))≤ϕi(Tj6(s1,s2,…,s|Tj6|)),i=0,1,…,n. The equality holds if and only if s2=⋯=s|Tj6|=0.

Lemma 3.5. For n≥|Tj3|(j=1,2,…,9),

Proof: (i) We have

Further by Lemma 2.3, T23(n-7,0,0,0,0,0,0)≽T13(n-7,0,0,0,0,0,0).

So (ii) and (iii) hold.     □

Lemma 3.6. For n≥|Tj4|(j=1,…,20),

Proof: