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 graph can be described as the sum of the absolute values of its eigenvalues. In this paper, we characterize tricyclic signed graphs with minimal energy.
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 total -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 graph is a pair , where is referred to the underlying graph of and 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.
Let be a cycle of a signed graph . The sign of , denoted by , is defined as . If or , then the signed cycle is said to be positive or negative, respectively. It is easy to see that , if it contains even number of negative edges, and otherwise. A signed graph is said to be balanced if all cycles in are positive; otherwise, it is said to be unbalanced. Throughout this paper, a positive cycle of order is denoted by , a negative cycle of order is denoted by , and we denote by a cycle of order which is either positive or negative. Let be the set of connected signed graphs on vertices and edges. In particular, for , 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 vertices are denoted by , , and , respectively. denotes the complete bipartite signed graph with parts of size and . The degree of a vertex in a signed graph is the number of edges that are incident to in denoted by . A basic figure is a graph in which each component is either an edge or a cycle.
The adjacency matrix of a signed graph , denoted by , is an matrix with , where is the number of edges between and . In the case that , namely, the sign of every edge in is positive, is exactly the classical adjacency matrix of . The characteristic polynomial of the adjacency matrix , , is called the characteristic polynomial of the signed graph and is denoted by . The eigenvalues of , denoted by , are called the eigenvalues of . Since is real symmetric, all eigenvalues of are real. The set of distinct eigenvalues of together with their multiplicities is called its spectrum.
The energy of a graph was defined as by Gutman [2] in 1978, where , , are the eigenvalues of . This concept was extended to signed graphs in [3] where the energy of a signed graph was defined as the sum of absolute values of eigenvalues of , namely, . 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 graph is balanced if and only if and its underlying graph have the same spectrum.
Suppose is a sign function. Switching by means forming a new signed graph whose underlying graph is the same as , but whose sign function is defined on an edge by . In other words, if we define a diagonal signature matrix with for each , then . If there exists a switching function such that , or equivalently , we say the signed graphs and 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 [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 graphs G with vertices and edges, the graphs with minimum energy are stars with additional edges all connected to the same vertex for 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 for in [13] and when for in [14]. In this paper, we consider the above conjecture for signed graphs, in the case for .
We also give the following definitions for signed graphs. Let be a signed star on vertices and be a set of signed graphs which are obtained from by adding signed edges joining one pendant vertex of to other pendant vertices. If a signed graph , then the signed graph is obtained by changing the sign of every edge in to the opposite. For , there are two switching classes in . One is positive containing cycle denoted by , and the other is negative containing cycle denoted by . For , there are three switching classes in . If the representative is balanced, we denote it by . If the representative contains , , and , then we denote it by , while has two and one . For , there are four switching classes in , respectively, expressed as , , , and . is balanced and has two positive cycles of length 3, , , and two negative cycles of length 4. Meanwhile, contains two negative cycles of length 3, , , and two negative cycles of length 4. And, is switching equivalent to with three negative cycles of length 3 and three positive cycles of length 4. For illustration, see Figure 1 for these four representatives in .
Let be the set of complete bipartite signed graphs formed by joining () signed pendant edges to a vertex of degree 4 of . There are three switching classes in . We use separately , , and as the representatives of the three switching classes. is balanced, contains three positive cycles of length 4 and three negative cycles of length 4, and contains two positive cycles of length 4 and four negative cycles of length 4. The three representatives are illustrated in Figure 2.
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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]). If is a signed graph with characteristic polynomialthenfor all , where is the set of all basic figures of of order , denotes the number of components of , denotes the set of all cycles of , and denotes the sign of cycle .
From Theorem 1, it is clear that the spectrum of a signed graph remains invariant by changing the signs of noncyclic edges.
Theorem 2 (see [13]). Let be a signed graph on vertices with characteristic polynomial . Then,
If we put and , then the above integral formula for signed graphs takes the form
It is easy to see that is a monotonously increasing function of . Let and be two signed graphs of order .(1)If and for all , then and we write .(2)If and for all , we see that and we write .(3)If and there exists some such that or , then we write . Clearly, is a quasiorder relation, that is, reflexive and transitive on coefficient. Therefore,
Note that the order of signed graphs does not have to be the same in a quasi-order comparison.
Let be the set of all connected tricyclic signed graphs with . It is straightforward to check that a signed graph has at least 3 signed cycles and at most 7 signed cycles. We classify into the following two categories.(i)Case 1. There exists a noncyclic path connecting one cycle with another denoted by . 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 . For illustration, see Figure 4.
If is a subset of the vertex set of the signed graph , then represents the signed graph obtained from by removing the vertices in . Let be a subset of the edge set of the signed graph ; then, by , we mean the signed graph obtained from by deleting the edges in . In addition, we refer to as a cut set of a signed graph if is disconnected.
Let be an complex matrix and be the conjugate transpose of . The singular values , , of a matrix are the positive square roots of the eigenvalues of . Since is a real symmetric matrix, and . For a partitioned matrix , where both and are square matrices, the following inequality about singular value can be find in [15].
It implies that . Proceeding exactly in a similar way as in [15], the following lemma illustrates the change in graph energy when a cut set is deleted.
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Lemma 2 (see [15]). If is a cut set of a signed graph , then . Moreover, if consists of a single edge, then .
The following three lemmas are needed in our paper.
Lemma 3 (see [2]). , with equality on left if and only if and equality on right if and only if .
Lemma 4 (see [13]). Among all unicyclic signed graphs with vertices, the signed graphs have the minimal energy. Moreover, for , the unicyclic signed graphs with the minimal energy are shown in Figure 5.
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Lemma 5 (see [14]). Among all bicyclic signed graphs with vertices, and have the minimal energy. For , shown in Figure 6 has the minimal energy. For , has the minimal energy and and have the second minimal energy. For , has the minimal energy.
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3. Main Results
In this section, we shall consider the signed tricyclic graphs in having the minimal energy.
Lemma 6. For , .
Proof. It is obvious that and because is switching equivalent to and is switching equivalent to . Therefore, to compare the energy of , for , it is enough to compare the energy of and . By Theorem 1, we haveHowever, we find that the coefficients of signed graph and are not quasi-order comparable. We will directly solve the integrals to compare the energy using Theorem 2, and we havePut and , and we get for and . Thus, .
Lemma 7. For , .
Proof. The characteristic polynomials of , , are given byClearly, . Therefore, .
Lemma 8 (see [9]). (i)For , .(ii)For , .
Lemma 9 (see [13]). For and , . The equality holds if and only if .
Theorem 3. (1) for , , .(2) for , .
Proof. (1)Note that and . It is enough to prove that Again, by Theorem 1, we have As , , we see that . So, .(2)The characteristic polynomial of is given byBy the characteristic polynomial of , we getNote that . In this case, we define and . It is easy to see that , . Also, ; therefore, has two positive and two negative zeros. We observe that , , , , , , , , , and . Therefore, has two positive and two negative zeros too. Let the positive zeros of be and . Using the fact that the energy of a signed graph is twice the sum of the positive eigenvalues, we havefor all . This completes the proof.
Theorem 4. If , , then .
Proof. Let be an edge of the noncyclic path in . Then, , where is a bicyclic signed graph on vertices and is a unicyclic signed graph on vertices. The following cases arise. (1) ; (2) ; (3) ; (4) ; (5) ; (6) ; (7) ; and (8) . Case (1): . By Lemmas 2 and 9 and Theorem 3, we have Case (2): . Proceeding similarly as in Case (1), we can prove that , where is the signed graph with the minimal energy for shown in Figure 5. Thus, we need to prove that . Note that . We have Case (3): . Note that . Then, Case (4): . Note that . Then, Case (5): . Note that . Then, Case (6): . Note that . Then, Case (7): . Note that . Then, Case (8): . Note that . Then,The theorem now holds.
Lemma 10. Let be a signed star of order and be a bicyclic signed graph of order , where , and . Then,
Proof. By Lemmas 3 and 5, we have . Then, we consider the following characteristic polynomial:Applying the integral formula given in Theorem 2, we havewhereSince and , we observe that , with equality if and only if .
Lemma 11. (i) for all .(ii) for all .(iii) for all .(iv) for all .(v) for all .
Proof. (i)Note that . Therefore, for all .(ii)We have for all .(iii)By direct calculations, we have verified for to 26, where . For , we consider the function We see , , , and for all . Therefore, has two positive zeros in the interval and , respectively. We have for all .(iv)Note that . We have for all .(v)We can easily verify for . When , we consider the functionWe see , , , and for all . Note that . Similar to (iii), we haveThe lemma holds.
Let such that there exists a cut set of size two whose deletion will result in a signed tree with order at least 3 and a bicyclic signed graph and be the set of all signed graphs .
Theorem 5. If with , then .
Proof. Let . Choosing the cut set , has two components, say and , where is a signed tree on vertices and is a bicyclic signed graph with vertices such that . By Lemma 3, . Then, the result follows by Lemmas 2, 5, 10, and 11.
By Theorem 5, we immediately have the following corollary.
Corollary 1. Let be a tricyclic signed graph in satisfying one of the following properties; then, .(i)There exists a path with , , , and , .(ii)There exists a vertex other than common vertices with .(iii)There exist two adjacent vertices other than common vertices denoted by with or .(iv)There exists a vertex not on the cycles with .(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 in , where is a subset of . Our results show that if , then which is a key to solving the conjecture (changed graphs to signed graphs) for and lays a foundation for future research of tricyclic signed graphs with minimal energy.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was funded by Shanxi Scholarship Council of China (no. 201901D211227).