On the Change of Distance Energy of Complete Bipartite Graph due to Edge Deletion

Abstract

The distance energy of a graph is defined as the sum of absolute values of distance eigenvalues of the graph. The distance energy of a graph plays an important role in many fields. By constructing a new polynomial, we transform a problem on the sum of the absolute values of the roots of a quadratic polynomial into a problem on the largest root of a cubic polynomial. Hence, we give a new and shorter proof on the change of distance energy of a complete bipartite graph due to edge deletion, which was given by Varghese et al.

1. Introduction

Let be a simple connected graph, whose vertex set is and edge set is . For any pair of vertices (), the distance is defined as the length of the shortest path between and . In particular, for any . The distance matrix of , denoted by , is the matrix with -entry being . The distance energy of is defined by the sum of absolute values of distance eigenvalues of . The distance energy of has been studied for several years in the literature, see [1–5] and the references therein.

It is not easy to compute the distance energy of general graphs. Thus, we are interested in finding the distance energy of some special graphs. For example, it has been proved in [6–8] thatwhere is a complete -partite graph with the size of each partition being at least 2. The distance energy of some graphs with diameter 2 is also determined in [9]. Another interesting problem is to study how the distance energy of a graph changes by deleting an edge. Zhou and Ilić [10] showed that the deletion of any edge increases the distance energy of a connected graph with unique positive distance eigenvalue whenever the resulting graph is still connected. Varghese et al. [11] proved that for any edge of , where is a complete bipartite graph with . Recently, Tian et al. [12] showed that the deletion of any edge increases the distance energy of some special complete multipartite graphs.

In this paper, we will give a new and shorter proof of the main result in [11] by transforming a problem on the sum of the absolute values of the roots of a quadratic polynomial into a problem on the largest root of a cubic polynomial.

2. Main Result

In this section, we will give a new and shorter proof on , which has been proved in [11]. Before giving our main proof, we need a result on distance eigenvalues of a graph.

Lemma 1 (see [13]). Let be a graph of order . If there exist vertices having the same neighborhood, then has distance eigenvalue with multiplicity at least .

Next, we introduce a concept about the equitable quotient matrix. Let be a symmetric matrix of order whose block form is as follows:where the blocks are matrices for any and . Let denote the sum of all entries in divided by the number of rows. Then, is called the quotient matrix of . If has a constant row sum for each pair (), then is called the equitable quotient matrix of . Haemers [14] obtained the relation between the spectra of and as follows.

Lemma 2 (see [14]). Let be the equitable quotient matrix of as defined above. Then, any eigenvalue of is a eigenvalue of .

Now, we are ready to give the main result of this paper.

Theorem 1. For integers and ,where is any edge of .

Proof. If or 5, one can easily confirm that holds; otherwise, . From the structure of the graph , by Lemma 1, we found that the distance eigenvalues of are with multiplicity . Note that, after relabeling the matrix , we rewrite as a block matrix in the following form:where be a matrix with all entries being 1. It is clear that each block of has a constant row sum. Thus, the equitable quotient matrix of isAfter simple calculation, we obtain the characteristic polynomial of asSince for any integer and , has four roots which are different from . Combining these results with Lemma 2, we conclude that has eigenvalue with multiplicity and the remaining four eigenvalues (say, ) are the roots of . By Descartes’ rule of signs with the fact that , one can easily check that . Now, let be the coefficients of . Note that and . Then, can be rewritten aswhere , , , and .

Claim 1. .

Proof of Claim. 1. From the given conditions, we haveOn the other hand, we haveAs , we obtainCombining (8)–(10), we can obtain that is the root of , whereFrom (7), it follows that has three possible values:Since , then is the largest root of . Using the fact that and , we havewhereAgain using the fact that , we obtainAs , it follows that . Combining the above results, we haveThis follows directly that . This completes the proof of Claim 1.
By Claim 1 with , we obtainHence, we haveThis completes the proof of this theorem.

Remark 1. It is worth mentioning that the method we used in the above proof would be a tool to compare the sum of the absolute values of the roots of a quadratic polynomial with a certain value.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first author was supported by the National Natural Science Foundation of China (Grant nos. 11901525 and 11801512) and Zhejiang Provincial Natural Science Foundation of China (LY20A010005).