Almost Borderenergetic Line Graphs

Introduction

In this manuscript, we consider graphs that are simple, undirected, and finite. Let G be a graph with an order of n, and let λ 1 ≥ λ 2 ≥ . . . ≥ λ n represent its eigenvalues, which are derived from its adjacency matrix A(G). The collection of these eigenvalues is referred to as the graph G's spectrum, denoted as Sp(G) (for a comprehensive exploration of spectral graph theory, see [14]). The spectrum of a graph is a powerful tool in understanding its structural and algebraic properties. It provides crucial insights into the behavior of various graph operations and algorithms.

The theory of graph spectra

The concept of graph spectra emerged from practical applications in Chemistry and Physics. The seminal paper [28] introduces graph spectra implicitly, and the initial mathematical investigation [6] is driven by the aim of solving the membrane vibration problem using approximate methods for partial differential equations. A prominent application in Chemistry involves the Hückel molecular orbital theory, further details available in references such as [1,7,12,19,22,23,25,38].

For mathematical point of view, approximately solving the membrane vibration problem, represented by a graph as a discrete model, involves considering the eigenvalues of the corresponding partial differential equation [12,Chapter 8]. The spectra of graphs, particularly those related to adjacency matrices, play a crucial role in statistical physics problems, such as the dimer problem, which explores the thermodynamic properties of diatomic molecules on a crystal surface, requiring the enumeration of non-overlapping dimer arrangements on a lattice [29,33,34].

Over the past decade, the practical applications of graph spectra in computer science have gained growing recognition, encompassing various domains such as internet technologies, pattern recognition, and computer vision. Graph eigenvalues, particularly in the context of expanders, play a pivotal role in solving various computer science problems, including communication networks, error-correcting codes, and memory optimization (for details, refer to [15,31,36]). The highest eigenvalue plays a pivotal role in modeling virus propagation in computer networks, where a lower maximum eigenvalue correlates with enhanced network resilience against virus spread [39,40]. Additionally, graph eigenvalues contribute to web search engine algorithms [4,30] and are employed in developing secure statistical databases, where eigenvalues, especially the least eigenvalue -2, are linked to compromise-free query collections (see [2,3,13,42]).

Furthermore, the theory of graph spectra intersects with linear algebra and combinatorial optimization. Combinatorial matrix theory, as presented in [5], utilizes graph spectra concepts for the foundation and development of matrix theory, covering determinants, linear algebraic equations, Perron-Frobenius theory, and more. Eigenvalues of graphs play a vital role in solving various combinatorial optimization problems, as demonstrated in the comprehensive expository article [32], where graph spectra are applied to address partition problems, ordering, stable sets and coloring, routing problems, embedding problems, and more. Additionally, the travelling salesman problem (TSP) incorporates semi-definite programming (SDP) and graph Laplacian eigenvalues to establish lower bounds and complexity indices, as outlined in [8][9][10]17].

Graph energy

Graph energy research traces its origins back to the 1940s when Erich Hückel introduced a method to approximate solutions to the Schrödinger Equation for a specific class of organic molecules known as unsaturated conjugated hydrocarbons. This method, commonly referred to as the Hückel molecular orbital (HMO) theory, is extensively expounded in relevant textbooks [7,37,41].

The connection between HMO and graph energy was first established by [26] and later reaffirmed by [16]. In the broader context of the HMO model, solving the eigenvalue-eigenvector problem of an approximate Hamiltonian matrix becomes essential. This matrix is represented as H = αI n + βA(G), where α and β are constants, I n is the unit matrix of order n, and A(G) is the adjacency matrix of a specific graph G corresponding to the carbon-atom skeleton of the conjugated molecule.

The energy levels (E j ) of the π-electrons are determined by the eigenvalues (λ j ) of the graph G through the simple relationship E j = α + βλ j . The molecular orbitals coincide with the corresponding eigenvectors (ψ j ) of the graph G. In the Hückel Molecular Orbital (HMO) approximation, the total energy of all π-electrons (E π ) is expressed as

, where g j is referred to as the occupation number, representing the count of π-electrons in alignment with the molecular orbital ψ j . Following a fundamental physical principle, g j is constrained to assume values of 0, 1, or 2.

In most chemically relevant scenarios, the occupation number g j follows:

Consequently, we have:

Since the sum of eigenvalues for all graphs equals zero, we can express the above equation as:

Gong et al. [21] introduced the concept of borderenergetic graphs, characterizing a graph G of order n as borderenergetic if its energy matches that of the complete graph K n , which is 2(n -1), i.e., if E(G) = 2(n -1). Some related findings on borderenergetic graphs can be explored in [18,21,24]. Conversely, graphs of order n with energy close to 2(n -1) are also noteworthy, particularly given the limited instances of borderenergetic graphs in specific scenarios, such as borderenergetic line graphs. We introduce the term almost borderenergetic graph to characterize a graph of order n where |E(G)-2(n-1)| < 1.

Our contributions

Consider a graph G with an edge set E. The line graph of G, denoted as L(G), is defined as a graph with a vertex set corresponding to E, where two vertices in L(G) are connected if and only if their corresponding edges in G share a common endpoint. Notably, the line graph of a regular graph is itself a regular graph. Specifically, if G is a regular graph of order n 0 with a degree of r 0 , then the line graph of G has an order of n 1 = 1 2 r 0 n 0 and a degree of r 1 = 2r 0 -2. Essential properties of line graphs are available in standard textbooks, such as [27].

Moreover, we utilize the concept of the join operation, denoted as G = G 1 ∇G 2 , for graphs G 1 and G 2 . This operation is defined as G = G 1 ∪ G 2 , where ∪ represents the union operation on two graphs, and G signifies the complement of the graph G. Similarly, the union of r copies of G is represented as

In this paper, we construct new borderenergetic graphs from the line graph of regular graphs. In particular, we prove conditions for the graph G = pG 1 ∪ qK k+1 consisting of p-copies of a k-regular graph G 1 and q-copies of the complete graph K k+1 to be borderenergetic, see Theorem 2.2. We present a methodology for constructing non-complete (2k -2)-regular borderenergetic graphs by utilizing smaller-order k-regular graphs. Next, we prove the conditions for the line graph of a strongly regular to be borderenergetic, see Theorem 3.3. These results show that conditions to be borderenergetic line graph are very strong and the exhaustive search gives few number of examples of borderenergetic line graphs. Therefore, in the second part of this paper, we study almost borderenergetic graphs whose energy differs from borderenergetic ones by 1. We first prove the energy of graph L((rK 1 )∇K 2 ) and show that {L((rK 1 )∇K 2 )|r ≥ 5} is an infinite family of connected almost borderenergetic graph in Theorem 4.3. In particular, we observe that the energy of L((rK 1 )∇K 2 ) approaches 4r as the parameter r increases, thus we define a family of asymptotic borderenergetic graphs.

The outline of the paper

The paper is organized as follows. In Section 2, we study the line graph of regular graph and related results about its energy. Then, we consider the line graph of the strongly regular graphs to be borderenergetic in Section 3. We study almost borderenergetic graphs in Section 4, and we conclude the paper in Section 5.

Borderenergetic line graphs from regular graphs

Firstly, we recall the following result from the spectral graph theory. We use the notation of s (t) to denote that the eigenvalue s occurs t-times in the spectrum of a graph. Theorem 2.1 [11,14] If λ 1 , λ 2 , . . . , λ r represent the adjacency eigenvalues of a k-regular graph G of order r, then the adjacency eigenvalues of L(G) can be expressed as follows:

.

By using Theorem 2.1, we present how to construct a non-complete (2k -2)-regular (k > 2) borderenergetic graphs by using some k-regular graphs of small order. Deng et al. [18] proved the cases that the complement of graph G in studied Theorem 2.2 is borderenergetic. We prove the conditions for which the line of G is borderenergetic.

Theorem 2.2 Consider G = pG 1 ∪ qK k+1 , a k-regular graph formed by combining pcopies of G 1 and q-copies of K k+1 , where G 1 is a k-regular graph with an order of r and has t positive eigenvalues, while the rest are less than 2 -k. If the following equation holds:

Then, L(G) is a non-complete borderenergetic graph.

. By Theorem 2.1, the spectrum of L(G) is as follows:

times.

Thus, the energy of the line graph of G can be found as

We define integer N as

By the above condition, we have

as desired. By Theorem 2.2, we arrive the following corollaries.

Corollary 2.3 Let G = pG 1 be a k-regular graph consisting of p-copies of G 1 , where G 1 is a k-regular graph of order r with t positive eigenvalues and others less than

Proof. For q = 0, we have the result from Theorem 2.2.

G L(G)

Figure 1. The graphs from Example 2.5

Corollary 2.4 Let G be a k-regular graph of order r with t positive eigenvalues and others less than

Proof. We obtain the result by taking p = 1 in Corollary 2.3. We remark that a (2k -2)-regular borderenergetic graph L(G) can be constructed from a k-regular graph G by Theorem 2.2.

Corollary 4.7 Let G r = (rK 1 )∇K 2 be a graph and L(G r ) be its line graph of order 2r +1 for r ∈ Z + . Then,

Proof. By Theorem 4.5, E(L(G r )) = 3r -4 + √ r 2 + 8r. On the other hand, E(K 2r+1 ) = 4r. Therefore, the limit of their ratio converges to 1 as r → ∞.

Remark 4.8 We note that E(L(G r )) reaches to E(K 2r+1 ) as r goes to infinity. Because of this property, we call Ω a family of asymptotic borderenergetic graphs. We tabulate L(G r ) for r = 1, 2, . . . , 20 in Table 1. It is seen in the table that the energy of L(G r ) converges to the energy of the complete graph K 2r+1 .

G

L(G) Example 2.7 Let G be a connected 3-regular graph with 10 vertices (Petersen Graph) besides L(G) be a connected 4-regular borderenergetic graph as given in Figure 3. Note that the energy of G satisfies

hence its line graph is borderenergetic by Corollary 2.4:

Borderenergetic line graphs from strongly regular graphs

A strongly regular graph SRG(n, k, λ, µ) is a graph of order n and valency k such that (i) it is non-complete or edgeless, (ii) every adjacent two vertices share λ common neighbours, and (iii) every non-adjacent two vertices share µ common neighbours. A SRG(n, k, λ, µ) has the following eigenvalues: k with multiplicity 1,

with multiplicity m 1 and θ 2 = (λ-µ)-

with multiplicity m 2 , where ∆ = (λ-µ) 2 +4(k-µ) and m 1 , m 2 such that m 1 + m 2 = n -1, m 1 θ 1 + m 2 θ 2 + k = 0, see [14,20]. By solving them,

Substituting θ 1 and θ 2 , we find

Firstly we give a necessary condition for SRG(n, k, λ, µ) to be borderenergetic.

then G is borderenergetic.

, we write

Thus the proof is completed by the equality E(G) = 2n -2.

Example 3.2

The graphs SRG(9, 4, 1, 2) and SRG(16, 5, 0, 2) are borderenergetic.

In the following, we will show how we can construct connected non-complete (2k -2)regular borderenergetic graphs from SRG(n, k, λ, µ).

Theorem 3.3 Let G be a strongly regular graph with parameters (n, k, λ, µ) and spectrum

. Then the line graph L(G) of G is borderenergetic if either of the following holds:

we have

and so

By the equality E(L(G)) = kn -2, we find that L(G) is borderenergetic. The case (ii) can be proven similarly.

Example 3.4

The line graphs of SRG(6, 3, 0, 3) and SRG(10, 3, 0, 1) are borderenergetic.

Almost borderenergetic line graphs

In this section, we study graphs of order n whose energy differ at most 1 from the energy of the complete graph of order n. We study this type of graphs because there are not enough number of examples of borderenergetic line graphs. In particular, as we have seen in Sections 2 and 3, there are strong conditions for the line graph from a regular graph to be borderenergetic, see Theorems 2.2 and 3.3. In addition, borderenergetic line graphs satisfying these theorems are rare. We can say that the only examples in small order are the ones given in Examples 2.5, 2.6, 2.7 and 3.4. Therefore, we find that constructing line graphs whose energy is very near to a borderenergetic graph is an interesting problem.

In this section, we will give an infinite family of line graphs with energies differ form borderenergetic graphs at most 1. We start with the formal definition of an almost borderenergetic graph.

The set of almost borderenergetic graphs is not empty. We below give an example of an almost borderenergetic graph by using line graph of join of two complete graphs. We now give an infinite set of almost borderenergetic graphs in Theorem 4.3. We note that this infinite set of almost borderenergetic connected and non-complete graphs is constructed by using line and join operators. Before giving the proof of this theorem, we state necessary lemmas below. Define I n and J n are the identity and all-one square matrix of dimension n. Lemma 4.4 Let G r = (rK 1 )∇K 2 be a graph for r ∈ Z + and complete graphs K 1 and K 2 of order 1 and 2, respectively.

i. Adjacency matrix

ii. Adjacency matrix A L of the line graph of G r is

iii. A L = X T X -2I 2r+1 , where X is a r + 2 × 2r + 1 matrix as follows

v. The set of eigenvalues of matrix XX T is

where α 1 and α 2 are the roots of f (x) = x 2 -(r + 4)x + 4.

Proof. The proof of (i)-(iv) follow by direct computation. Here, we give the proof of item (v). The characteristic polynomial of XX T is computed as follows:

Therefore, we see that the spectrum of XX T is obtained as given in Lemma (v). Proof. Let G r = (rK 1 )∇K 2 be a graph of order r + 2 for r ≥ 5. Then, the characteristic polynomial P L(Gr) of the adjacency matrix of its line graph L(G r ) is computed below:

Therefore, by Lemma 4.4 (v), the spectrum of matrix L(G r ) is

where α 1 and α 2 are the roots of f (x) = x 2 -(r + 4)x + 4. Thus, we get the energy of the line graph L(G r ) of G r computed by using (1) as follows

which completes the proof. We now give the proof of Theorem 4.3. Proof. To show that the graph L(G r ) of order 2r + 1 is almost borderenergetic, we need to prove that |E(L(G r )) -4r| < 1 or |3r -4 + √ r 2 + 8r -4r| < 1, which follows by Theorem 4.5. It is easy to see that the later holds for r ≥ 5. We now present in the following theorem an asymptotic property of the energy of the family Ω given above.

Conclusion

We proved the necessary condition for the line graph of the union of some regular graphs to be borderenergetic. In particular, these graphs form a family of non-complete borderenergetic (2k -2)-regular graph obtained from the line graph of a k-regular graph for k ≥ 2. Next, we presented the necessary condition for the line graph of a strongly regular graph to be borderenergetic. Then, we explicitly obtained the spectrum of the line graph L(rK 1 ∇K 2 ), and by using this result, we constructed an infinite set of almost borderenergetic graphs. We note that it is a research problem to consider other special graph families and their line graphs in order to be borderenergetic. Constructing other examples of asymptotic borderenergetic graph families would be a future work. Generalizing the definition of almost borderenergetic graphs to k-almost case would be studied, where this type of graphs have energy differing from the complete graph by k.