# Research on Extreme Signed Graphs with Minimal Energy in Tricyclic Signed GraphsS(n, n + 2)

Authors: Yajing Wang; Yubin Gao
Journal: Complexity
 (2020)

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## Abstract

A signed graph is acquired by attaching a sign to each edge of a simple graph, and the signed graphs have been widely used as significant computer models in the study of complex systems. The energy of a signed graphS can be described as the sum of the absolute values of its eigenvalues. In this paper, we characterize tricyclic signed graphs with minimal energy.


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## Body

## 1. Introduction

There are various complex systems, such as the society, biology, and information technology systems. In these complex systems, oppositional relationships are universal and can be described abstractly as signed networks. The research of signed networks is of great significance for the accurate understanding of complex systems and their application, which has attracted more and more attention. For example, combining positive and negative information can allow a more accurate identification of topics in sensor networks and make more effective recommendations on social networks. Since 1953 [1], Harary put forward signed graphs for a quantitative analysis of the balance of all parties in social relations. Experimental results show that the use of signed networks is effective to achieve the prediction of major social events. Since then, the signed graph has become an important mathematical model to describe the topological structure of signed networks. Meanwhile, the signed graph model also enriches the research content of spectral graph theory. Then, many research studies about unsigned graphs are extended to signed graphs.In the field of chemical research, chemists have found that the energy of the molecular graphs of most hydrocarbon molecules is linearly related to the totalπ-electrical energy of the molecule. Due to this significant application in physical and chemical sciences, the research on energy of graphs has attracted the attention of many chemical and mathematical researchers. The goal of this paper is to investigate the tricyclic signed graphs with minimal energy.Formally, a signed graphS is a pair G,σ, where G=V,E is referred to the underlying graph of S and σ:E⟶+,− is the sign function or signature. The signed graphs mentioned in this paper have simple underlying graphs. We draw the positive edge with a plain line and the negative edge with a dotted line. A Signed graph is referred to as a generalization of a graph because a graph can be considered as a signed graph where each edge is positive.LetC be a cycle of a signed graph S. The sign of C, denoted by sgnC, is defined as sgnC=∏e∈ECσe. If sgnC=+ or sgnC=−, then the signed cycle is said to be positive or negative, respectively. It is easy to see that sgnC=+, if it contains even number of negative edges, and sgnC=− otherwise. A signed graph S is said to be balanced if all cycles in S are positive; otherwise, it is said to be unbalanced. Throughout this paper, a positive cycle of order n is denoted by Cn+, a negative cycle of order n is denoted by Cn−, and we denote by Cn a cycle of order n which is either positive or negative. Let Sn,m be the set of connected signed graphs on n vertices and m edges. In particular, for m=n,n+1,n+2, Sn,m is the set of connected unicyclic signed graphs, bicyclic signed graphs, and tricyclic signed graphs, respectively. The signed tree, the signed path, and the signed star on n vertices are denoted by Tn, Pn, and Sn, respectively. Kn1,n2 denotes the complete bipartite signed graph with parts of size n1 and n2. The degree of a vertex v in a signed graph S is the number of edges that are incident to v in S denoted by dv. A basic figure is a graph in which each component is either an edge or a cycle.The adjacency matrix of a signed graphS, denoted by AS=aijσ, is an n×n matrix with aijσ=σvivjaij, where aij is the number of edges between vi and vj. In the case that σ=+, namely, the sign of every edge in S is positive, AS=aij+ is exactly the classical adjacency matrix of G. The characteristic polynomial of the adjacency matrix AS, detxI−AS, is called the characteristic polynomial of the signed graph S and is denoted by ϕS,x. The eigenvalues of AS, denoted by x1,x2,…,xn, are called the eigenvalues of S. Since AS is real symmetric, all eigenvalues of S are real. The set of distinct eigenvalues of S together with their multiplicities is called its spectrum.The energy of a graphG was defined as EG=∑i=1nλi by Gutman [2] in 1978, where λi, i=1,2,…,n, are the eigenvalues of G. This concept was extended to signed graphs in [3] where the energy of a signed graph S was defined as the sum of absolute values of eigenvalues of S, namely, ES=∑j=1nxj. Furthermore, this concept of energy has been extended to digraphs in [4] and signed digraphs in [5].The spectral criterion for the balance of signed graphs given by Acharya [6] is as follows.Lemma 1 (see [6]).
A signed graphS is balanced if and only if S and its underlying graph have the same spectrum.Supposeθ:VG⟶+,− is a sign function. Switching S by θ means forming a new signed graph Sθ=G,σθ whose underlying graph is the same as G, but whose sign function is defined on an edge uv by σθuv=θuσuvθv. In other words, if we define a diagonal signature matrix Dθ with dv=θv for each v∈VG, then ASθ=DθASDθ. If there exists a switching function θ such that S1=S2θ, or equivalently AS2=DθAS1Dθ, we say the signed graphs S1 and S2 are switching equivalent. Switching equivalent is an equivalence relation on the sign of a fixed underlying graph. An equivalence class is called a switching class. Switching a signed graph does not change the sign or the balance of cycles in S [7]. That is to say, if two signed graphs are switching equivalent, then they have the same set of positive cycles, and they are either both balanced or both unbalanced. Furthermore, switching equivalent preserves the spectrum. Thus, as long as spectrum are concerned, we use a single signed graph to represent a switching class. Many results on the minimal energy have been obtained for various classes of graphs. The authors of [8] gave the following conjecture.Conjecture 1 (see [8]).
Among all connected graphsG with n≥6 vertices and n−1≤m≤2n−2 edges, the graphs with minimum energy are stars with m−n+1 additional edges all connected to the same vertex for m≤n+⌊n−7/2⌋ and bipartite graphs with two vertices on one side, one of which is connected to all vertices on the other side otherwise.The conjecture is true for unicyclic, bicyclic, and tricyclic graphs [9–12]. A signed analogue of this conjecture is true when m=n for n≥6 in [13] and when m=n+1 for n≥12 in [14]. In this paper, we consider the above conjecture for signed graphs, in the case m=n+2 for n≥14.We also give the following definitions for signed graphs. LetSn be a signed star on n vertices and Sn,m be a set of signed graphs which are obtained from Sn by adding m−n+1 signed edges joining one pendant vertex of Sn to other m−n+1 pendant vertices. If a signed graph Sn,m∈Sn,m, then the signed graph −Sn,m∈Sn,m is obtained by changing the sign of every edge in Sn,m to the opposite. For m=n, there are two switching classes in Sn,n. One is positive containing cycle C3+ denoted by Sn,n1, and the other is negative containing cycle C3− denoted by Sn,n2. For m=n+1, there are three switching classes in Sn,n+1. If the representative is balanced, we denote it by Sn,n+11. If the representative contains C3+, C3−, and C4−, then we denote it by Sn,n+12, while Sn,n+13 has two C3− and one C4+. For m=n+2, there are four switching classes in Sn,n+2, respectively, expressed as Sn,n+21, Sn,n+22, Sn,n+23, and Sn,n+24. Sn,n+21 is balanced and Sn,n+22 has two positive cycles of length 3, C3−, C4+, and two negative cycles of length 4. Meanwhile, Sn,n+23 contains two negative cycles of length 3, C3+, C4+, and two negative cycles of length 4. And, Sn,n+24 is switching equivalent to −Sn,n+21 with three negative cycles of length 3 and three positive cycles of length 4. For illustration, see Figure 1 for these four representatives in Sn,n+2.LetK2,n−2 be the set of complete bipartite signed graphs formed by joining n−6 (n≥6) signed pendant edges to a vertex of degree 4 of K2,4. There are three switching classes in K2,n−2. We use separately K2,n−21, K2,n−22, and K2,n−23 as the representatives of the three switching classes. K2,n−21 is balanced, K2,n−22 contains three positive cycles of length 4 and three negative cycles of length 4, and K2,n−23 contains two positive cycles of length 4 and four negative cycles of length 4. The three representatives are illustrated in Figure 2.Figure 1
The four representatives of switching classes in tricyclic signed graphsSn,n+2. (a) Sn,n+21. (b) Sn,n+22. (c) Sn,n+23. (d) Sn,n+24.
(a)(b)(c)(d)Figure 2
Bipartite tricyclic signed graphs with two vertices on one side. (a)K2,n−21. (b) K2,n−22. (c) K2,n−23.
(a)(b)(c)


## 2. Preliminaries

In this section, we shall give some definitions and notations and state some results, which will be needed to prove our main results.Theorem 1 (see [6]).
IfS is a signed graph with characteristic polynomial(1)ϕS,x=xn+a1Sxn−1+⋯+an−1Sx+anS,then(2)ajS=∑L∈φj−1pL2cL∏Z∈cLsZ,for all j=1,2,…,n, where φj is the set of all basic figures L of S of order j, pL denotes the number of components of L, cL denotes the set of all cycles of L, and sZ denotes the sign of cycle Z.From Theorem1, it is clear that the spectrum of a signed graph remains invariant by changing the signs of noncyclic edges.Theorem 2 (see [13]).
LetS be a signed graph on n vertices with characteristic polynomial ϕS,x=xn+a1Sxn−1+⋯+an−1Sx+anS. Then,(3)ES=12π∫−∞+∞1x2log∑j=0n/2−1ja2jSx2j2+∑j=0n/2−1ja2j+1Sx2j+12dx.If we putb2jS=−1ja2jS and b2j+1S=−1ja2j+1S, then the above integral formula for signed graphs takes the form(4)ES=12π∫−∞+∞1x2log∑j=0n/2b2jSx2j2+∑j=0n/2b2j+1Sx2j+12dx.It is easy to see thatES is a monotonously increasing function of bjS. Let S1 and S2 be two signed graphs of order n.(1)
Ifb2jS1=b2jS2 and b2j+1S1=b2j+1S2 for all j≥0, then ES1=ES2 and we write S1∼S2.(2)
Ifb2jS1≥b2jS2 and b2j+1S1≥b2j+1S2 for all j≥0, we see that ES1≥ES2 and we write S1≽S2.(3)
IfS1≽S2 and there exists some j0 such that b2j0S1>b2j0S2 or b2j0+1S1>b2j0+1S2, then we write S1≻S2. Clearly, ≽ is a quasiorder relation, that is, reflexive and transitive on coefficient. Therefore,(5)S1≻S2⟹ES1>ES2.Note that the order of signed graphs does not have to be the same in a quasi-order comparison.LetSn,n+2 be the set of all connected tricyclic signed graphs with m=n+2. It is straightforward to check that a signed graph S∈Sn,n+2 has at least 3 signed cycles and at most 7 signed cycles. We classify Sn,n+2 into the following two categories.(i)
Case 1. There exists a noncyclic path P connecting one cycle with another denoted by S1n,n+2. For illustration, see Figure 3, where we have not shown noncyclic edges which can be present.(ii)
Case 2. There exist two edges’ disjoint paths between any two vertices which are on some cycles, denoted by S2n,n+2. For illustration, see Figure 4.IfV1 is a subset of the vertex set V of the signed graph S, then S−V1 represents the signed graph obtained from S by removing the vertices in V1. Let E1 be a subset of the edge set E of the signed graph S; then, by S−E1, we mean the signed graph obtained from S by deleting the edges in E1. In addition, we refer to E1 as a cut set of a signed graph S if S−E1 is disconnected.LetM be an m×n complex matrix and M∗ be the conjugate transpose of M. The singular values σ1M, σ2M,…, σnM of a matrix M are the positive square roots of the eigenvalues of MM∗. Since M=AS is a real symmetric matrix, ASAS∗=AS2 and ES=∑inσiAS. For a partitioned matrix A=BXYC, where both B and C are square matrices, the following inequality about singular value can be find in [15].(6)∑iσiB+∑iσiC≤∑iσiA.It implies thatEB+EC≤EA. Proceeding exactly in a similar way as in [15], the following lemma illustrates the change in graph energy when a cut set is deleted.Figure 3
Seven underlying graphs of tricyclic signed graphs in case 1.
(a)(b)(c)(d)(e)(f)(g)Figure 4
Eight underlying graphs of tricyclic signed graphs in case 2.
(a)(b)(c)(d)(e)(f)(g)(h)Lemma 2 (see [15]).
IfE1 is a cut set of a signed graph S, then ES−E1≤ES. Moreover, if E1 consists of a single edge, then ES−E1<ES.The following three lemmas are needed in our paper.Lemma 3 (see [2]).
ESn≤ETn≤EPn, with equality on left if and only if Tn=Sn and equality on right if and only if Tn=Pn.Lemma 4 (see [13]).
Among all unicyclic signed graphs withn≥6 vertices, the signed graphs Sn,n have the minimal energy. Moreover, for n=3,4,5, the unicyclic signed graphs with the minimal energy are shown in Figure 5.Figure 5
Unicyclic signed graphs with minimal energy. (a)C3+. (b) C3−. (c) C4+. (d) S0. (e) Sn,n1.
(a)(b)(c)(d)(e)Lemma 5 (see [14]).
Among all bicyclic signed graphs withn≥8 vertices, Sn,n+11 and Sn,n+13 have the minimal energy. For n=6,7, Bn,n+11 shown in Figure 6 has the minimal energy. For n=5, B5,61 has the minimal energy and S5,61 and S5,63 have the second minimal energy. For n=4, S4,51 has the minimal energy.Figure 6
Bicyclic signed graphs with minimal energy. (a)Sn,n+11. (b) Sn,n+13. (c) Bn,n+11.
(a)(b)(c)


## 3. Main Results

In this section, we shall consider the signed tricyclic graphs inSn,n+2n≥14 having the minimal energy.Lemma 6.
Forn≥7, ESn,n+21=ESn,n+24<ESn,n+22=ESn,n+23.Proof.
It is obvious thatESn,n+21=ESn,n+24 and ESn,n+22=ESn,n+23 because Sn,n+21 is switching equivalent to −Sn,n+24 and Sn,n+22 is switching equivalent to −Sn,n+23. Therefore, to compare the energy of Sn,n+2r, for r=1,2,3,4, it is enough to compare the energy of Sn,n+21 and Sn,n+23. By Theorem 1, we have(7)ϕSn,n+21=xn−n+2xn−2−6xn−3+3n−15xn−4,ϕSn,n+23=xn−n+2xn−2+2xn−3+3n−7xn−4.
However, we find that the coefficients of signed graphSn,n+21 and Sn,n+23 are not quasi-order comparable. We will directly solve the integrals to compare the energy using Theorem 2, and we have(8)ESn,n+23−ESn,n+21=1π∫0+∞1x2log1+n+2x2+3n−7x42+4x61+n+2x2+3n−15x42+36x6dx.
Putαx=1+n+2x2+3n−7x42+4x6 and βx=1+n+2x2+3n−15x42+36x6, and we get αx−βx=163n−11x8+16nx6+16x4>0 for n≥7 and x>0. Thus, ESn,n+23>ESn,n+21.Lemma 7.
Forn≥7, EK2,n−21<EK2,n−22<EK2,n−23.Proof.
The characteristic polynomials ofϕK2,n−2r, r=1,2,3, are given by(9)ϕK2,n−21=xn−n+2xn−2+4n−24xn−4,ϕK2,n−22=xn−n+2xn−2+4n−12xn−4,ϕK2,n−23=xn−n+2xn−2+4n−8xn−4.
Clearly,K2,n−21≺K2,n−22≺K2,n−23. Therefore, EK2,n−21<EK2,n−22<EK2,n−23.Lemma 8 (see [9]).
(i)
Forn≥11, ESn,n+21<EKn,n−21.(ii)
For7≤n≤10, EKn,n−21<ESn,n+21.Lemma 9 (see [13]).
Forn1,n2≥6 and n=n1+n2, ESn1,n11∪Sn2,n21≥ESn−6,n−61∪S6,61. The equality holds if and only if n1,n2∈6,n−6.Theorem 3.
(1)
ESn,n+1r1>ESn,nr2 for r1=1,2,3, r2=1,2, n≥6.(2)
ESn−6,n−6r∪S6,6r>EK2,n−23 for r=1,2, n≥12.Proof.
(1)
Note thatSn,n1∼Sn,n2 and Sn,n+11∼Sn,n+13≺Sn,n+12. It is enough to prove that(10)ESn,n+11>ESn,n1.Again, by Theorem1, we have(11)ϕSn,n+11=xn−n+1xn−2−4xn−3+2n−4xn−4,ϕSn,n1=xn−nxn−2−2xn−3+n−3xn−4.Asn≥6, 2n−4>n−3>0, we see that Sn,n+11≻Sn,n1. So, ESn,n+11>ESn,n1.(2)
The characteristic polynomial ofK2,n−23 is given by(12)ϕK2,n−23=xn−n+2xn−2+4n−8xn−4.
By the characteristic polynomial ofSn,n1, we get(13)ϕSn−6,n−61=xn−6−n−6xn−8−2xn−9+n−9xn−10.
Note thatES6,61≈6.1722>6. In this case, we define αx=x4−n+2x2+4n−8 and βx=x4−n−6x2−2x+n−9. It is easy to see that α2=0, αn−2=0. Also, α−x=αx; therefore, αx has two positive and two negative zeros. We observe that β0>0, β−n>0, β−2<0, β−1=0, β−n>0, β−2<0, β−1=0, β1/2>0, βn−7<0, and βn−6>0. Therefore, βx has two positive and two negative zeros too. Let the positive zeros of βx be b1 and b2. Using the fact that the energy of a signed graph is twice the sum of the positive eigenvalues, we have(14)ESn−6,n−6r∪S6,6r=ESn−6,n−61∪S6,61=ESn−6,n−61+ES6,61>2b1+b2+6>212+n−7+6=7+2n−7>22+n−2=EK2,n−23,for all n≥12. This completes the proof.Theorem 4.
IfS∈S1n,n+2, n≥14, then ES>EK2,n−23.Proof.
Lete1 be an edge of the noncyclic path P in S∈S1n,n+2. Then, S−e1=S1∪S2, where S1 is a bicyclic signed graph on n1 vertices and S2 is a unicyclic signed graph on n2 vertices. The following cases arise. (1) n1≥8,n2≥6; (2) n1≥9,n2=5; (3) n1≥10,n2=4; (4) n1≥11,n2=3; (5) n1=7,n2≥7; (6) n1=6,n2≥8; (7) n1=5,n2≥9; and (8) n1=4,n2≥10.Case (1):n1≥8,n2≥6. By Lemmas 2 and 9 and Theorem 3, we have(15)ES>ES−e=ES1∪S2=ES1+ES2≥ESn1,n1+11+ESn2,n21>ESn1,n11+ESn2,n21≥ESn−6,n−61+ES6,61>EK2,n−23.Case (2):n1≥9,n2=5. Proceeding similarly as in Case (1), we can prove that ES>ESn−5,n−5+ES0, where S0 is the signed graph with the minimal energy for n=5 shown in Figure 5. Thus, we need to prove that ESn−5,n−5+ES0>EK2,n−23. Note that ES0≈5.5960>5.5. We have(16)ESn−5,n−51+ES0>212+n−6+5.5=6.5+2n−6>22+n−2=EK2,n−23.Case (3):n1≥10,n2=4. Note that EC4+=4. Then,(17)ESn−4,n−41+EC4+>212+n−5+4=5+2n−5>22+n−2=EK2,n−23.Case (4):n1≥11,n2=3. Note that EC3+=4. Then,(18)ESn−3,n−31+EC3+>212+n−4+4=5+2n−4>22+n−2=EK2,n−23.Case (5):n1=7,n2≥7. Note that EB7,81≈7.1831>7. Then,(19)EB7,81+ESn−7,n−71>7+212+n−8=8+2n−8>22+n−2=EK2,n−23.Case (6):n1=6,n2≥8. Note that EB6,71≈6.4696>6. Then,(20)EB6,71+ESn−6,n−61>6+212+n−7=7+2n−7>22+n−2=EK2,n−23.Case (7):n1=5,n2≥9. Note that EB5,61=26. Then,(21)EB5,61+ESn−5,n−51>4.9+212+n−6=5.9+2n−6>22+n−2=EK2,n−23.Case (8):n1=4,n2≥10. Note that ES4,51≈5.1231>5. Then,(22)ES4,51+ESn−4,n−41>5+212+n−5=6+2n−5>22+n−2=EK2,n−23.
The theorem now holds.Lemma 10.
LetSn1 be a signed star of order n1 and Sn2,n2+1 be a bicyclic signed graph of order n2, where n1≥3, n2≥8 and n1+n2=n. Then,(23)ESn1∪Sn2,n2+1r≥ES3∪Sn2,n2+11≥ES3∪Sn−3,n−21.Proof.
By Lemmas3 and 5, we have ESn1∪Sn2,n2+1r≥ES3∪Sn2,n2+11. Then, we consider the following characteristic polynomial:(24)ϕSn1,xϕSn2,n2+11,x=xn1−n1−1xn1−2xn2−n2+1xn2−2−4xn2−3+2n2−4xn2−4=xn−nxn−2−4xn−3+n1n2+n−9xn−4+4n1−1xn−5−2n1−1n2−4xn−6.
Applying the integral formula given in Theorem2, we have(25)ESn1∪Sn2,n2+11=1π∫0+∞1x2logfxdx≥1π∫0+∞1x2loggxdx=ES3∪Sn−3,n−21,where(26)fx=1+nx2+n1n2+n−9x4+2n1−1n2−4x62+4x3+4n1−1x52,(27)gx=1+nx2+4n−18x4+4n−7x62+4x3+8x52.
Sincen1≥3 and n1+n2=n, we observe that n1n2≥3n−3, with equality if and only if n1,n2∈3,n−3.Lemma 11.
(i)
ESn−7∪B7,81>EK2,n−23 for all n≥14.(ii)
ESn−6∪B6,71>EK2,n−23 for all n≥14.(iii)
ESn−5∪B5,61>ESn,n+21 for all n≥14.(iv)
ESn−4∪S4,51>EK2,n−23 for all n≥14.(v)
ES3∪Sn−3,n−21>EK2,n−23 for all n≥14.Proof.
(i)
Note thatspecSn=±n−1,0n−2. Therefore,(28)ESn−7∪B7,81>2n−8+7>22+n−2=EK2,n−23,for alln≥14.(ii)
We have(29)ESn−6∪B6,71>2n−7+6>22+n−2=EK2,n−23,for alln≥14.(iii)
By direct calculations, we have verifiedESn,n+21<2n−6+26 for n=14 to 26, where EB5,61=26. For n≥27, we consider the function(30)αx=x4−n+2x2−6x+3n−15.We seeα2>0, α3<0, αn−1<0, and αn+1>0 for all n≥27. Therefore, αx has two positive zeros in the interval 2,3 and n−1,n+1, respectively. We have(31)ESn−5∪B5,61=2n−6+26>23+n+1>ESn,n+21,for alln≥27.(iv)
Note thatES4,51>5. We have(32)ESn−5∪S4,51>2n−5+5>22+n−2=EK2,n−23,for alln≥14.(v)
We can easily verifyES3+ESn−3,n−21>EK2,n−23 for n=14. When n≥15, we consider the function(33)αx=x4−n−2x2−4x+2n−7.
We seeα1≥0, α2<0, αn−4<0, and αn−2>0 for all n≥15. Note that ES3=22. Similar to (iii), we have(34)ES3+ESn−3,n−21>22+21+n−4=22+2+2n−4>4+2n−2=EK2,n−23.
The lemma holds.
LetH∈S2n,n+2 such that there exists a cut set E1 of size two whose deletion will result in a signed tree with order at least 3 and a bicyclic signed graph and ℋ be the set of all signed graphs H.Theorem 5.
IfH∈ℋ with n≥14, then ES>ESn,n+21.Proof.
LetH∈ℋ. Choosing the cut set E1=e1,e2, H−e1,e2 has two components, say H1 and H2, where H1 is a signed tree on n1≥3 vertices and H2 is a bicyclic signed graph with n2 vertices such that n1+n2=n≥14. By Lemma 3, EH1≥ESn1. Then, the result follows by Lemmas 2, 5, 10, and 11.
By Theorem5, we immediately have the following corollary.Corollary 1.
LetS be a tricyclic signed graph in S2n,n+2 satisfying one of the following properties; then, ES>ESn,n+21.(i)
There exists a pathv0,v1,…,vk with k≥4, dv0≥3, dvk≥3, and dvi=2, i=1,…,k−1.(ii)
There exists a vertexu other than common vertices with du≥4.(iii)
There exist two adjacent vertices other than common vertices denoted byw,z with dw≥3 or dz≥3.(iv)
There exists a vertexv not on the cycles with dv≥3.(v)
There exists a pendant path of length at least 3 with one endpoint on the cycles or there exists a pendant path of length at least 2 with one endpoint not on the cycles.


## 4. Conclusions

The spectral properties of signed graphs are of great significance in the application of spectral graph theory as well as the topology of complex systems. Generally, graph energy is an important combinatorial spectral form. We characterized the structure of tricyclic signed graphs inℋ, where ℋ is a subset of S2n,n+2. Our results show that if S∈S1n,n+2∪ℋ, then ES>ESn,n+21 which is a key to solving the conjecture (changed graphs to signed graphs) for m=n+2 and lays a foundation for future research of tricyclic signed graphs with minimal energy.



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