Abstract
Let be a graph with vertices. For every real , write for the matrix , where and denote the adjacency matrix and the degree matrix of , respectively. The collection of eigenvalues of together with multiplicities are called the -spectrum of . A graph is said to be determined by its -spectrum if all graphs having the same -spectrum as are isomorphic to . In this paper, we show that some joins are determined by their -spectra for .
1. Introduction
We use to denote a simple graph with vertex set and edge set . The degree of a vertex is denoted by . For a subgraph of , let denote the subgraph obtained from by deleting the edges of . Let and denote, respectively, the numbers of -cycles and -vertex paths in . Let denote the number of triangles containing the vertex of . Let be the union of two graphs and which have no common vertices. For any positive integer , let denote be the union of disjoint copies of graph . The join of two disjoint graphs and , denoted by , is the graph obtained by joining each vertex of to each vertex of . For convenience, the complete graph, path, cycle and star on vertices are denoted by , , , and , respectively.
Let and denote, respectively, the adjacency matrix and degree matrix of . For every real , write for the matrix . Note that and , where is the signless Laplacian matrix of . The polynomialis called -characteristic polynomial, where is the identity matrix of order . The theory of -characteristic polynomial of a graph is well elaborated [1–8].
The -spectrum of is a collection of roots of together with multiplicities. Two graphs are said to be -cospectral if they have the same -spectrum. A graph is called an -DS graph if it is determined by its -spectrum, meaning that there exists no other graph that is nonisomorphic to it but -cospectral with it.
It is interesting to characterize which graph is determined by some graph spectrum [9–11]. The problem was raised by Günthard and Primas [12] in 1956 with motivations from chemistry. In recent years, although many graphs have been proved to be DS graphs, the problem of determining DS graphs is still far from being completely solved [13, 14]. Recently, Lin et al. [15] considered the problem which graph is determined by its -spectrum? And they gave some characterizing properties of -spectrum and proposed the following problem.
Problem 1. Characterizing graphs determined by their -spectra such that is also determined by their -spectra for or .
Liu and Lu [16] discussed the problem which join graph is determined by its -spectrum? And they pointed out the following problem.
Problem 2. Prove or disprove that is determined by its -spectrum for .
In this paper, we focus on Problem 1 above, and we prove that some join graphs are -DS graphs. Furthermore, we also give a special solution for Problem 2. The rest of this paper is organized as follows. In Section 2, we present some characterizing properties of the -spectrum of graphs and give the formula to compute in , where is a subgraph of with edges. In Section 3, we give a solution for Problem 1.
2. Preliminaries
Let denote the set of graphs each of which is obtained from by removing five or fewer edges. For , there exist exactly 45 nonisomorphic graphs each of which is obtained from by removing five or fewer edges [17]. These graphs are labeled by , and illustrated in Figure 1. Checking the structure of , we know that , where is a graph obtained form deleting some edges, and , e.g., .
Cámara and Haemers [18] discussed the problem which is determined by its -spectrum. And they gave the following result.
Theorem 1 (see [18]). Let be a graph with vertices. Then, is -DS graph.
Lemma 1 (see [19]). Let be a graph with edges and let . Then,
By Lemma 1, the number of triangles of some is calculated [17], see Table 1.
Lemma 2 (see [17]). Let be a graph with edges and let . Then,
By Lemma 2, the number of quadrangles of some is calculated [17], see Table 2.
Using the Principle of Inclusion-Exclusion, we can obtain the following result.
Lemma 3. Let be a graph with edges and let . Let , and let be an endpoint of edges in . Then,
Proof. Let . Let denote the set of triangles of containing and . Thus, there exists exactly triangles containing in . By the Inclusion-Exclusion Principle, we haveFor any edge , if is an endpoint of , then there exists triangles containing . Otherwise, there exists triangles containing . So, . For any given and , if is a common endpoint of and , then there exists triangles containing and . Otherwise, there exists triangles containing and in , where is a path which is origin endpoint and is the number of vertices with length 2 to . Thus, . Since any two edges in edges induce a triangle, . By the above arguments, we arrive in equation (4).
Lemma 4 (see [20] and [5]). Let be a graph with vertices and edges, and let be the degree sequence of . Suppose that . Then,(i)(ii)(iii)(iv)(v)
For convenience, by Lemmas 3 and 4, we calculate the value of some graphs in , see Table 3.
Lemma 5 (see [21]). Let and be two graphs with vertices. For , if and are -cospectral, then the following statements hold:(i).(ii).(iii)If is -regular, then is -regular. Suppose that and are the degree sequences of and , respectively. If and are -cospectral with , then(iv).(v).
Lemma 6 (see [21]). The complete graph is determined by its -spectrum.
Lemma 7 (see [21]). The graph is determined by its -spectrum, where and .
By Lemma 7, we can obtain a corollary as follows.
Corollary 1. Graphs , , , , and are determined by their -spectra, where .
The M-coronal of an square matrix , denoted by , is defined to be the sum of the entries of the matrix , that is,where denotes the column vector of size n with all the entries equal to one and means the transpose of ([22, 23]).
Lemma 8 (see [16]). If is an arbitrary graph and and are Q-cospectral graphs with , then and are -cospectral.
By Lemma 8, we obtain directly the following corollary.
Corollary 2. If is an arbitrary graph and and are -cospectral graphs with , then and are -cospectral.
Lemma 9. The each of following holds:(i) and are -cospectral, where (ii) and are -cospectral, where
Proof. Directly calculating the signless Laplacian polynomials of and yield . Furthermore, by simple computations, we have . By Corollary 2, it is easy to see that the results in Lemma 9 hold.
By Lemma 9, we obtain some -cospectral mates in .
Corollary 3. The following results hold:(i)Graphs and are -cospectral(ii)Graphs and are -cospectral(iii)Graphs and are -cospectral(iv)Graphs and are -cospectral
Remark 1. By Corollaries 1 and 3, we know that is a -DS graph, and and are -cospectral. These results answer the special case of Problem 2.
3. Main Results
In this section, we show that all graphs in are determined by their -spectra.
Theorem 2. Graphs and are -DS graphs, where .
Proof. The result follows from Lemma 5 and Corollary 1.
Theorem 3. Let G be a graph obtained from by deleting three edges, and then is determined by the -spectra when .
Proof. Checking Figure 1, we know that is isomorphic to one of {, , , , }. Directly computing yields , , and . By Lemma 4 (iv) and Table 1, we haveSolving equationwe have , , or . This implies that for .
By Corollaries 1 and 3 (i) and Lemma 7 (i), (ii), and (v), the result in Theorem 3 holds.
Remark 2. By the proof of Theorem 3, it can be known that , , and are determined by their -spectra.
Lemma 10. Each of the following holds:(i)Graphs and are not -cospectral, where (ii)Graphs and are not -cospectral, where (iii)Graphs , , and are not pairwise -cospectral, where
Proof. (i)By Lemma 4 (iv) and Table 1, we have . Solving equation we obtain , 0 or 0. It implies that and are not -cospectral, when .(ii)By Lemma 4 () and Tables 1–3, we obtain that Solving equation we have , 1, , or . This indicates that and are not -cospectral when .(iii)Similarly, by Lemma 4 (iv) and Table 1, we obtain that Solving equation we obtain , 1 or 1. By the roots of equations (8), (9), and (13), we know that , , and are not pairwise -cospectral when .
Theorem 4. Graphs , , , , , , , , , , and are determined by their -spectra, respectively, where .
Proof. By simple computations, we obtain that , , , , and .
By Corollaries 1 and 3 (ii) and Lemmas 5 and 10, graphs , , , , , , , , , , and are -DS graphs, where .
Lemma 11. Each of the following holds:(i)Graphs and are not -cospectral, where (ii)Graphs and are not -cospectral, where (iii)Graphs and are not -cospectral, where (iv)Graphs , , , , and are not pairwise -cospectral, where .(v)Graphs , , , and are not pairwise -cospectral, where (vi)Graphs , , , and are not pairwise -cospectral, where (vii)Graphs , , , and are not pairwise -cospectral, where
Proof. (i)By Lemma 4 (v) and Tables 1–3, we have By the roots of equations (11), we know that and not -cospectral when . (ii) By Lemma 4 (iv) and Table 1, we have . Solving equation we have , , or . It implies that and are not -cospectral when .(iii)Similarly, by Lemma 4 (iv) and Table 1, we have . By the roots of equation (15), we know that and are not -cospectral, where .(iv)Analogously, by Lemma 4 (iv) and (v) and Tables 1–3, we obtain that Solving equation we obtain , 1, 1, or . By the roots of equations (8), (9), (11), (13), and (17), we obtain that , , , , and are not pairwise when .(v)Similarly, by Lemma 4 (iv) and (v) and Tables 1–3, we obtain that Solving equation we obtain or . By the roots of equations (9), (11), (13), and (19), we obtain that , , , and are not pairwise when .(vi)By Lemma 4 (iv) and (v) and Tables 1–3, we have Solving equation we obtain , , or . By the roots of equations (8), (11), (13), and (21), we obtain that , , , and are not -cospectral when .(vii)Finally, by Lemma 4 (iv) and (v) and Tables 1–3, we have By the roots of equations (8), (9), (11), and (13), we obtain that , , , and are not -cospectral when .
Theorem 5. Graphs , , , , , , , , , , , , , , , , , , , , , , , , and are, respectively, determined by their -spectra, where when .
Proof. By simple computations, we have that , , , , , , , , and .
By Corollaries 1 and 3 (iii) and (iv) and Lemmas 5 and 11, graphs , , , , , , , , , , , , , , , , , , , , , , , , , and are determined by their -DS graphs, respectively, where .
By Corollary 1 and Theorems 2–5, directly yields the following result.
Theorem 6. Let be a graph with vertices. is determined by its -spectrum, where .
Remark 3. By Theorems 2–4, we know that almost complete graphs are determined by their -spectra, where , each is a join. Thus, these results is a solution of Problem 1. Motivated by these results, we pose the following two questions.
Question 1. Prove or disprove that and are -cospectral, where .
Question 2. Prove or disprove that and are -cospectral, where .
Data Availability
Data from previous studies were used to support this study. They are cited at relevant places within the text as references.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (no. 11761056), Natural Science Foundation of Qinghai Province (no. 2020-ZJ-920), and Scientific Research Innovation Team in Qinghai Nationalities University.