# SkyMath: Technical Report

Liu Yang, Haihua Yang, Wenjun Cheng, Lei Lin, Chenxia Li, Yifu Chen, Lunan
Liu,
Jianfei Pan, Tianwen Wei, Biye Li, Liang Zhao, Lijie Wang, Bo Zhu, Guoliang
Li,
Xuejie Wu, Xilin Luo, Rui Hu†

Kunlun Inc.
<EMAIL_ADDRESS>

###### Abstract

Large language models (LLMs) have shown great potential to solve varieties of
natural language processing (NLP) tasks, including mathematical reasoning. In
this work, we present SkyMath, a large language model for mathematics with 13
billion parameters. By applying self-compare fine-tuning, we have enhanced
mathematical reasoning abilities of Skywork-13B-Base remarkably. On GSM8K,
SkyMath outperforms all known open-source models of similar size and has
established a new SOTA performance. On dataset MATH and out-of-domain dataset
CMath, SkyMath also achieves a high accuracy rate, showing remarkable
generalizability to varieties of math problems.

##  1 Introduction

Nowadays, Large language models (LLMS) are increasingly being applied to all
kinds of complex tasks, including content creation (Gardner et al., 2023;
Zhang et al., 2023a; Zhu et al., 2023; Dai et al., 2023; Liu et al., 2023a),
code generation (Nijkamp et al., 2023; Wang et al., 2023d; Fried et al., 2023;
Chen et al., 2021; Li et al., 2023; Zheng et al., 2023b), multi-turn
conversations (Thoppilan et al., 2022; Touvron et al., 2023b; Bai et al.,
2023; Yang et al., 2023a; Zeng et al., 2023; Shu et al., 2023), mathematical
reasoning (Azerbayev et al., 2023; Yue et al., 2023; Yu et al., 2023; Luo et
al., 2023a; Qiao et al., 2023; Chern et al., 2023; Fu et al., 2023a), and
knowledge-based question answering (Cui et al., 2023; Wu et al., 2023; Yang et
al., 2023b; Wang et al., 2023a; Zhu & Wang, 2023), and they have the potential
to revolutionize the fields of natural language processing and natural
language understanding (OpenAI, 2022, 2023). Moreover, compared to traditional
AI methods, LLMs gain unparalleled advantages in these landscapes. Generative
Artificial Intelligence (GenAI) fueled by LLMs is just around the corner.

Despite their impressive capabilities, LLMs come with a series of challenges
and issues. Mathematical reasoning is one of subfields worth exploring. In the
context of evaluating the reasoning capabilities of LLMs, complex mathematical
reasoning serves as a crucial benchmark. Moreover, mathematical reasoning
tasks are considered to be one of the major gaps between closed-source models
like ChatGPT (OpenAI, 2022) or GPT4 (OpenAI, 2023) and open-source models like
LLaMA (Touvron et al., 2023a, b). To achieve advanced mathematical reasoning
capabilities, numerous open-source models have made attempts: Kuaishou
introduces the KwaiYiiMath by applying Supervised Fine-Tuning (SFT) and
Reinforced Learning from Human Feedback (RLHF) and constructed a small-scale
Chinese primary school mathematics test set (named KMath) (Fu et al., 2023a);
Microsoft presentes wizardMath by applying their proposed Reinforcement
Learning from Evol-Instruct Feedback (RLEIF) method to the domain of math (Luo
et al., 2023a); Xiang et al. introduce MAmmoTH by training the model on their
self-developed dataset MathInstruct (Yue et al., 2023); Zhangir et al.
presente LLEMMA by continue pretraining Code Llama on Proof-Pile-2, a mixture
of scientific papers, web data containing mathematics, and mathematical code
(Azerbayev et al., 2023); Yu et al. (Yu et al., 2023) proposes MetaMath
trained on a new dataset called MetaMathQA which is build by bootstrapping
mathematical questions by rewriting the question from multiple perspectives.
Despite the great success, most existing open-source LLMs are still far away
from satisfactory for solving mathematical problems due to the complex
reasoning procedures, and preform poorly on GSM8K and MATH when compared to
open-sourced models.

To bridge this gap, we propose SkyMath, a finetuned language model that
specializes in mathematical reasoning, by applying our improved data
augmentation techniques and SFT process to the finetuning of Skywork-13B-Base.
The main process consists of three parts, as follows: 1. Finetuning the
Skywork-13B-Base model on open-source datasets; 2. Constructing a new dataset
for math by improved data augmentation techniques; 3. Reconstructing math
dataset by applying our proposed self-check techniques. Experimental results
show that SkyMath outperforms many open-source models in similar sizes on two
mathematical benchmarks, namly GSM8k (Cobbe et al., 2021) and MATH (Hendrycks
et al., 2021).

The paper is structured as follows. Section 2 provides an overview of related
work including LLMs and LLMs for mathematics. Section 3 introduces the
methodology of Skywork-13B-Math including sample construction methods and
process of supervised fine-tuning . Then, the experimental validations and
comparisons are made in Section 4 and Section 5. The conclusion is drawn in
Section 6.

##  2 Related Work

#### Large Language Models

Recently, LLMs have achieved substantial progress in natural language
processing tasks, benefiting from high-quality, diverse textual data, and
enormous model parameter counts. Researchers have found that LLMs perform
better on downstream tasks (Kaplan et al., 2020). LLMs, which pretrained on
extensive textual data and fine-tuned for specific tasks, demonstrate
remarkable capabilities in numerous natural language processing tasks compared
to smaller models (Devlin et al., 2019; Wei et al., 2022b). The significant
capabilities of large models mainly lie in three aspects: 1. In-context
learning. 2. Instruction following. 3. Step-by-step reasoning (Zhao et al.,
2023). The GPT-3 (Brown et al., 2020), a language model that has tens of
billions parameter counts, exhibits significant performance improvements in
few-shot, one-shot, and zero-shot learning tasks through in-context learning.
However, the effectiveness of in-context learning capability depends on
different downstream tasks. The open-source model Llama2 (Touvron et al.,
2023b), with a pre-trained context length of up to 4k, demonstrates robust
contextual understanding in tasks such as summarization, multi-turn dialogues,
and reading comprehension. LLMs handle different downstream tasks efficiently,
but sometimes these pre-trained models still struggle to comprehend human
instructions (Ouyang et al., 2022). It is highly challenging for LLMs to
understand and follow specific instructions in the complex natural language
tasks. To overcome the challenge, Instruction Tuning (IT) (Zhang et al.,
2023b) and Chain of Thought (CoT) (Wei et al., 2022c) have been proposed.
Specifically, IT aims to utilize the constructed pairs whose formats are
[instruction, output] to fine-tune the pre-trained language model. Moreover,
the adoption of multitask instruction blending in fine-tuning has demonstrated
the effectiveness of IT techniques in unseen tasks (Sanh et al., 2022; Ouyang
et al., 2022; Wei et al., 2022a). Numerous studies show that high-quality,
diverse instructions can effectively improve the performance of LLMs in
natural language tasks (Wang et al., 2023c; Zhou et al., 2023; Wei et al.,
2022a; Cao et al., 2023). CoT is a method that aims to improve the performance
of LLMs by providing detailed reasoning processes as training inputs. By
enabling the model to follow a step-by-step process, CoT helps LLMs to break
down complex problems into smaller ones and accumulate small victories to
achieve greater success (Wei et al., 2022c). This approach has led to
significant improvements in the reasoning capabilities of LLMs, especially in
mathematical reasoning and decision-making tasks. The success of CoT and IT
technologies in Self-Supervised Fine-Tuning (SFT) has opened up new
possibilities for enhancing the performance of LLMs. Numerous models,
including InstructonGPT (Ouyang et al., 2022), BLOOMZ (Muennighoff et al.,
2023), WizardLM (Xu et al., 2023), ChatGLM2 (Du et al., 2022), and Vicuna
(Chiang et al., 2023), have acquired powerful reasoning abilities, allowing
them to perform complex tasks with remarkable accuracy. In addition, by
training LLMs on data specific to a particular domain, such as medical or
legal, it is possible to obtain models that excel in that domain. Models
obtained in this way include InstructDial (Gupta et al., 2022), Radiology-GPT
(Liu et al., 2023b), Goat (Liu & Low, 2023), WizardCoder (Luo et al., 2023b).

#### Large Language Models for Mathematical reasoning

Mathematical reasoning is one of the essential abilities required for LLMs. To
enhance mathematical reasoning capabilities of base LLMs, a group of
researchers conducted a comprehensive study. Based on CoT, a series of works
are carried out to optimize the reasoning paths (Wang et al., 2023b; Fu et
al., 2023b; Huang et al., 2022). Wang et al. introduces Self-Consistency,
employing the results of multiple reasoning paths to enhance the accuracy of
inference through consistent answers (Wang et al., 2023b). Fu et al. proposes
Complexity-based CoT, achieving improved performance in mathematical reasoning
tasks through a CoT involving multiple complex reasoning steps (Fu et al.,
2023b). In terms of prompting engineering, Zheng et al. combines CoT to
generate answers, utilizing the answer from the previous step as a prompt for
the next step, gradually guiding LLMs to generate correct answers (Zheng et
al., 2023a). Luo et al. proposes reinforced evol-instruct, a method that
combines evol-instruct with reinforcement learning to enhance the reasoning
performance of LLMs (Luo et al., 2023a). Madaan et al. introduces Self-Refine,
which a single LLM serves simultaneously as a generator, refiner, and feedback
provider, iteratively refining itself to enhance its capabilities in
mathematical reasoning (Madaan et al., 2023). Yu et al. constructs a new
dataset, MetaMathQA, through data augmentation of rewriting of mathematical
problems from multiple perspectives (Yu et al., 2023). Fu et al. enhances the
mathematical reasoning capabilities of LLMs through a combination of
supervised fine-tuning and reinforcement learning with human feedback (RLHF)
(Fu et al., 2023a). Based on Cot and Program of Thoughts (PoT) (Chen et al.,
2022), Yue et al. constructs a new dataset namly MathInstruct and uses it in
SFT. PoT utilizes an external interpreter (e.g., Python interpreter) to
compute answers for complex mathematical problems (Yue et al., 2023).
Azerbayev et al. continues pretraining on Proof-Pile-2, a dataset containing
mathematical web data and mathematical code, and uses PoT to obtain an
accurate answer (Azerbayev et al., 2023).

##  3 Method

In this section, we introduce SkyMath in detail. As shown in Figure 1, our
method mainly contains two steps:
1\. Instruction boosting.
2\. Self-compare fine-tuning.

Figure 1: The Overview Architecture of SkyMath

###  3.1 Instruction Boosting

Previous work has shown math instructions with various complexities could make
remarkable effect on math LLMs training (Luo et al., 2023a). Therefore, the
first step we do is constructing a dataset of both high quality and diversity.
1\. We first collect a math dataset from different sources, including
different levels, in both Chinese and English.
2\. Then inspired by WizardLM (Xu et al., 2023) and MetaMath (Yu et al.,
2023), we adapt instruction boosting, namely 1) concretizing, 2) adding
constraints, 3) deepening and 4) rephrasing, to the question augmentation
process.
3\. We use LLMs to generate responses for augmented questions.
4\. Correctness check.
By now, we have got a dataset of high complexity.

###  3.2 Self-compare Fine-tuning

Progressive-Hint Prompting (PHP) (Zheng et al., 2023a) enables multiple
interactions between users and LLMs by using previously generated answers as
hints to progressively guide LLMs toward the correct answers. Inspired by
this, we believe introducing previous answers to the training process also has
an effect. We hope the model can compare its previous answers with ground
truth and correct its specific errors through training.

Figure 2: Self-compare Fine-tuning

As shown in Figure 2, self-compare fine-tuning contains four steps:
1\. For each question, ask the LLM to give an answer.
2\. 2. Construct self-compare prompts, as shown in Figure 3.
3\. Combine the data with the origin dataset.
4\. Fine-tuning.
Like human beings, we believe LLMs tend to make different mistakes as their
abilities improve. Therefore, in practice we divide the origin dataset into
several sub-datasets thus we can repeat self-compare fine-tuning more than
once.

Figure 3: Self-compare Prompts

##  4 Experiments

###  4.1 Evaluation Benchmarks

We mainly evaluate SkyMath on two popular mathematical reasoning benchmarks:
GSM8k (Cobbe et al., 2021) and MATH (Hendrycks et al., 2021). In order to see
the performance of SkyMath on the out-of-domain dataset and in chinese, we
also evaluate our model on CMath (Wei et al., 2023), for completeness.

GSM8k contains 7,473 training data and 1,319 test data, mainly on grade school
level English math problems. Each problem takes between 2 and 8 steps to
solve, and solutions primarily involve performing a sequence of elementary
calculations using basic arithmetic operations.

MATH is much more challenging. It contains 7,500 training data and 5,000 test
data, spans seven subjects including Prealgebra, Algebra, Number Theory,
Counting and Probability, Geometry, Intermediate Algebra, and Precalculus.

CMath is a Chinese Elementary School Math Word Problems (CMATH) dataset, which
contains 1.7k elementary school-level math word problems with detailed
annotations, sourced from actual Chinese workbooks and exams. We use it as an
out-of-domain dataset.

###  4.2 Model and Baselines

We use SkyWork-13B as base model and correspondingly choose models of the same
size and has already been open source for comparison. Therefore we choose
LLaMA1 (Touvron et al., 2023a), LLaMA2 (Touvron et al., 2023a), BaiChuan1
(Yang et al., 2023a), BaiChuan2 (Yang et al., 2023a), WizardMath (Luo et al.,
2023a), GAIRMath-Abel (Chern et al., 2023), and MetaMath (Yu et al., 2023) as
baselines.

##  5 Main Results

Table 1: Results of pass@1 (%) on GSM8k, MATH and CMath Model | #Params | GSM8K | Math | CMath
---|---|---|---|---
LLaMA1 (Touvron et al., 2023a) | 13B | 17.8 | 3.9 | -
LLaMA2 (Touvron et al., 2023a) | 13B | 28.7 | 3.9 | -
BaiChuan1 (Yang et al., 2023a) | 13B | 26.76 | 4.84 | 51.33
BaiChuan2 (Yang et al., 2023a) | 13B | 52.77 | 10.08 | -
WizardMath (Luo et al., 2023a) | 13B | 63.9 | 14.0 | 50.83
GAIRMath-Abel (Chern et al., 2023) | 13B | 66.41 | 17.34 | -
MetaMath (Yu et al., 2023) | 13B | 72.3 | 22.4 | -
SkyMath | 13B | 72.33 | 16.98 | 77.27

Evaluating results are shown in Table 1. SkyMath outperforms all baselines on
GSM8K, thus we have established a new SOTA performance across open-source LLMs
of similar size. On the MATH dataset, which is highly challenging, SkyMath
also achieves a high accuracy rate. Meanwhile, on the out-of-domain dataset
CMath, SkyMath’s performance shows remarkable generalizability to both unseen
math problems and Chinese math problems.

##  6 Conclusion and Future Work

This paper introduces SkyMath, a mathematics model fine-tuned with self-
compare. Evaluating results show SkyMath achieves SOTA performance across all
existing open-source LLMs of similar size on GSM8K and shows remarkable
generalizability to out-of-domain math problems. Considering we didn’t use
tools or reward models, the result is even more difficult.
Future Work. Although our model achieves impressive performance on GSM8K, we
still fall far behind models like GPT-4. In the future, we will explore more
methods to improve the abilities of our model.

## References

  * Azerbayev et al. (2023) Zhangir Azerbayev, Hailey Schoelkopf, Keiran Paster, Marco Dos Santos, Stephen McAleer, Albert Q. Jiang, Jia Deng, Stella Biderman, and Sean Welleck.  Llemma: An open language model for mathematics.  _arXiv preprint arXiv:2310.06786_ , 2023.
  * Bai et al. (2023) Jinze Bai, Shuai Bai, Yunfei Chu, Zeyu Cui, Kai Dang, Xiaodong Deng, Yang Fan, Wenbin Ge, Yu Han, Fei Huang, Binyuan Hui, Luo Ji, Mei Li, Junyang Lin, Runji Lin, Dayiheng Liu, Gao Liu, Chengqiang Lu, Keming Lu, Jianxin Ma, Rui Men, Xingzhang Ren, Xuancheng Ren, Chuanqi Tan, Sinan Tan, Jianhong Tu, Peng Wang, Shijie Wang, Wei Wang, Shengguang Wu, Benfeng Xu, Jin Xu, An Yang, Hao Yang, Jian Yang, Shusheng Yang, Yang Yao, Bowen Yu, Hongyi Yuan, Zheng Yuan, Jianwei Zhang, Xingxuan Zhang, Yichang Zhang, Zhenru Zhang, Chang Zhou, Jingren Zhou, Xiaohuan Zhou, and Tianhang Zhu.  Qwen technical report, 2023.
  * Brown et al. (2020) Tom B. Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel M. Ziegler, Jeffrey Wu, Clemens Winter, Christopher Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei.  Language models are few-shot learners.  In Hugo Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin (eds.), _Proceedings of the 33th Annual Conference on Neural Information Processing Systems 2020_ , 2020.
  * Cao et al. (2023) Yihan Cao, Yanbin Kang, and Lichao Sun.  Instruction mining: High-quality instruction data selection for large language models.  _arXiv preprint arXiv:2307.06290_ , 2023.
  * Chen et al. (2021) Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Pondé de Oliveira Pinto, Jared Kaplan, Harrison Edwards, Yuri Burda, Nicholas Joseph, Greg Brockman, Alex Ray, Raul Puri, Gretchen Krueger, Michael Petrov, Heidy Khlaaf, Girish Sastry, Pamela Mishkin, Brooke Chan, Scott Gray, Nick Ryder, Mikhail Pavlov, Alethea Power, Lukasz Kaiser, Mohammad Bavarian, Clemens Winter, Philippe Tillet, Felipe Petroski Such, Dave Cummings, Matthias Plappert, Fotios Chantzis, Elizabeth Barnes, Ariel Herbert-Voss, William Hebgen Guss, Alex Nichol, Alex Paino, Nikolas Tezak, Jie Tang, Igor Babuschkin, Suchir Balaji, Shantanu Jain, William Saunders, Christopher Hesse, Andrew N. Carr, Jan Leike, Joshua Achiam, Vedant Misra, Evan Morikawa, Alec Radford, Matthew Knight, Miles Brundage, Mira Murati, Katie Mayer, Peter Welinder, Bob McGrew, Dario Amodei, Sam McCandlish, Ilya Sutskever, and Wojciech Zaremba.  Evaluating large language models trained on code.  _arXiv preprint arXiv:2107.03374_ , 2021.
  * Chen et al. (2022) Wenhu Chen, Xueguang Ma, Xinyi Wang, and William W. Cohen.  Program of thoughts prompting: Disentangling computation from reasoning for numerical reasoning tasks.  _arXiv preprint arXiv:2211.12588_ , 2022.
  * Chern et al. (2023) Ethan Chern, Haoyang Zou, Xuefeng Li, Jiewen Hu, Kehua Feng, Junlong Li, and Pengfei Liu.  Generative ai for math: Abel.  https://github.com/GAIR-NLP/abel, 2023.
  * Chiang et al. (2023) Wei-Lin Chiang, Zhuohan Li, Zi Lin, Ying Sheng, Zhanghao Wu, Hao Zhang, Lianmin Zheng, Siyuan Zhuang, Yonghao Zhuang, Joseph E. Gonzalez, Ion Stoica, and Eric P. Xing.  Vicuna: An open-source chatbot impressing gpt-4 with 90%* chatgpt quality, March 2023.  URL https://lmsys.org/blog/2023-03-30-vicuna/.
  * Cobbe et al. (2021) Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, and John Schulman.  Training verifiers to solve math word problems.  _arXiv preprint arXiv:2110.14168_ , 2021.
  * Cui et al. (2023) Jiaxi Cui, Zongjian Li, Yang Yan, Bohua Chen, and Li Yuan.  Chatlaw: Open-source legal large language model with integrated external knowledge bases.  _arXiv preprint arXiv:2306.16092_ , 2023.
  * Dai et al. (2023) Wenliang Dai, Junnan Li, Dongxu Li, Anthony Meng Huat Tiong, Junqi Zhao, Weisheng Wang, Boyang Li, Pascale Fung, and Steven C. H. Hoi.  Instructblip: Towards general-purpose vision-language models with instruction tuning.  _arXiv preprint arXiv:2305.06500_ , 2023.
  * Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.  BERT: pre-training of deep bidirectional transformers for language understanding.  In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pp. 4171–4186, 2019.
  * Du et al. (2022) Zhengxiao Du, Yujie Qian, Xiao Liu, Ming Ding, Jiezhong Qiu, Zhilin Yang, and Jie Tang.  GLM: general language model pretraining with autoregressive blank infilling.  In Smaranda Muresan, Preslav Nakov, and Aline Villavicencio (eds.), _Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics_ , pp. 320–335. Association for Computational Linguistics, 2022.
  * Fried et al. (2023) Daniel Fried, Armen Aghajanyan, Jessy Lin, Sida Wang, Eric Wallace, Freda Shi, Ruiqi Zhong, Scott Yih, Luke Zettlemoyer, and Mike Lewis.  Incoder: A generative model for code infilling and synthesis.  In _Proceedings of the 11th International Conference on Learning Representations_. OpenReview.net, 2023.
  * Fu et al. (2023a) Jiayi Fu, Lei Lin, Xiaoyang Gao, Pengli Liu, Zhengzong Chen, Zhirui Yang, Shengnan Zhang, Xue Zheng, Yan Li, Yuliang Liu, Xucheng Ye, Yiqiao Liao, Chao Liao, Bin Chen, Chengru Song, Junchen Wan, Zijia Lin, Fuzheng Zhang, Zhongyuan Wang, Di Zhang, and Kun Gai.  Kwaiyiimath: Technical report, 2023a.
  * Fu et al. (2023b) Yao Fu, Hao Peng, Ashish Sabharwal, Peter Clark, and Tushar Khot.  Complexity-based prompting for multi-step reasoning.  In _Proceedings of the 11th International Conference on Learning Representations_. OpenReview.net, 2023b.
  * Gardner et al. (2023) Josh Gardner, Simon Durand, Daniel Stoller, and Rachel M. Bittner.  Llark: A multimodal foundation model for music, 2023.
  * Gupta et al. (2022) Prakhar Gupta, Cathy Jiao, Yi-Ting Yeh, Shikib Mehri, Maxine Eskénazi, and Jeffrey P. Bigham.  Instructdial: Improving zero and few-shot generalization in dialogue through instruction tuning.  In Yoav Goldberg, Zornitsa Kozareva, and Yue Zhang (eds.), _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , pp. 505–525. Association for Computational Linguistics, 2022.
  * Hendrycks et al. (2021) Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob Steinhardt.  Measuring mathematical problem solving with the MATH dataset.  In Joaquin Vanschoren and Sai-Kit Yeung (eds.), _Proceedings of the Neural Information Processing Systems Track on Datasets and Benchmarks 1_ , 2021.
  * Huang et al. (2022) Jiaxin Huang, Shixiang Shane Gu, Le Hou, Yuexin Wu, Xuezhi Wang, Hongkun Yu, and Jiawei Han.  Large language models can self-improve.  _arXiv preprint arXiv:2210.11610_ , 2022.
  * Kaplan et al. (2020) Jared Kaplan, Sam McCandlish, Tom Henighan, Tom B. Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei.  Scaling laws for neural language models.  _arXiv preprint arXiv:2001.08361_ , 2020.
  * Li et al. (2023) Raymond Li, Loubna Ben Allal, Yangtian Zi, Niklas Muennighoff, Denis Kocetkov, Chenghao Mou, Marc Marone, Christopher Akiki, Jia Li, Jenny Chim, Qian Liu, Evgenii Zheltonozhskii, Terry Yue Zhuo, Thomas Wang, Olivier Dehaene, Mishig Davaadorj, Joel Lamy-Poirier, João Monteiro, Oleh Shliazhko, Nicolas Gontier, Nicholas Meade, Armel Zebaze, Ming-Ho Yee, Logesh Kumar Umapathi, Jian Zhu, Benjamin Lipkin, Muhtasham Oblokulov, Zhiruo Wang, Rudra Murthy V, Jason Stillerman, Siva Sankalp Patel, Dmitry Abulkhanov, Marco Zocca, Manan Dey, Zhihan Zhang, Nour Moustafa-Fahmy, Urvashi Bhattacharyya, Wenhao Yu, Swayam Singh, Sasha Luccioni, Paulo Villegas, Maxim Kunakov, Fedor Zhdanov, Manuel Romero, Tony Lee, Nadav Timor, Jennifer Ding, Claire Schlesinger, Hailey Schoelkopf, Jan Ebert, Tri Dao, Mayank Mishra, Alex Gu, Jennifer Robinson, Carolyn Jane Anderson, Brendan Dolan-Gavitt, Danish Contractor, Siva Reddy, Daniel Fried, Dzmitry Bahdanau, Yacine Jernite, Carlos Muñoz Ferrandis, Sean Hughes, Thomas Wolf, Arjun Guha, Leandro von Werra, and Harm de Vries.  Starcoder: may the source be with you!  _arXiv preprint arXiv:2305.06161_ , 2023.
  * Liu et al. (2023a) Haotian Liu, Chunyuan Li, Qingyang Wu, and Yong Jae Lee.  Visual instruction tuning.  _arXiv preprint arXiv:2304.08485_ , 2023a.
  * Liu & Low (2023) Tiedong Liu and Bryan Kian Hsiang Low.  Goat: Fine-tuned llama outperforms GPT-4 on arithmetic tasks.  _arXiv preprint arXiv:2305.14201_ , 2023.
  * Liu et al. (2023b) Zhengliang Liu, Aoxiao Zhong, Yiwei Li, Longtao Yang, Chao Ju, Zihao Wu, Chong Ma, Peng Shu, Cheng Chen, Sekeun Kim, Haixing Dai, Lin Zhao, Dajiang Zhu, Jun Liu, Wei Liu, Dinggang Shen, Xiang Li, Quanzheng Li, and Tianming Liu.  Radiology-gpt: A large language model for radiology.  _arXiv preprint arXiv:2306.08666_ , 2023b.
  * Luo et al. (2023a) Haipeng Luo, Qingfeng Sun, Can Xu, Pu Zhao, Jianguang Lou, Chongyang Tao, Xiubo Geng, Qingwei Lin, Shifeng Chen, and Dongmei Zhang.  Wizardmath: Empowering mathematical reasoning for large language models via reinforced evol-instruct.  _arXiv preprint arXiv:2308.09583_ , 2023a.
  * Luo et al. (2023b) Ziyang Luo, Can Xu, Pu Zhao, Qingfeng Sun, Xiubo Geng, Wenxiang Hu, Chongyang Tao, Jing Ma, Qingwei Lin, and Daxin Jiang.  Wizardcoder: Empowering code large language models with evol-instruct.  _arXiv preprint arXiv:2306.08568_ , 2023b.
  * Madaan et al. (2023) Aman Madaan, Niket Tandon, Prakhar Gupta, Skyler Hallinan, Luyu Gao, Sarah Wiegreffe, Uri Alon, Nouha Dziri, Shrimai Prabhumoye, Yiming Yang, Sean Welleck, Bodhisattwa Prasad Majumder, Shashank Gupta, Amir Yazdanbakhsh, and Peter Clark.  Self-refine: Iterative refinement with self-feedback.  _arXiv preprint arXiv:2303.17651_ , 2023.
  * Muennighoff et al. (2023) Niklas Muennighoff, Thomas Wang, Lintang Sutawika, Adam Roberts, Stella Biderman, Teven Le Scao, M. Saiful Bari, Sheng Shen, Zheng Xin Yong, Hailey Schoelkopf, Xiangru Tang, Dragomir Radev, Alham Fikri Aji, Khalid Almubarak, Samuel Albanie, Zaid Alyafeai, Albert Webson, Edward Raff, and Colin Raffel.  Crosslingual generalization through multitask finetuning.  In Anna Rogers, Jordan L. Boyd-Graber, and Naoaki Okazaki (eds.), _Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), ACL 2023, Toronto, Canada, July 9-14, 2023_ , pp. 15991–16111. Association for Computational Linguistics, 2023.
  * Nijkamp et al. (2023) Erik Nijkamp, Bo Pang, Hiroaki Hayashi, Lifu Tu, Huan Wang, Yingbo Zhou, Silvio Savarese, and Caiming Xiong.  Codegen: An open large language model for code with multi-turn program synthesis.  In _Proceedings of the 11th International Conference on Learning Representations_. OpenReview.net, 2023.
  * OpenAI (2022) OpenAI.  Openai: Introducing chatgpt.  2022\.
  * OpenAI (2023) OpenAI.  Gpt-4 technical report.  2023\.
  * Ouyang et al. (2022) Long Ouyang, Jeffrey Wu, Xu Jiang, Diogo Almeida, Carroll L. Wainwright, Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, Alex Ray, John Schulman, Jacob Hilton, Fraser Kelton, Luke Miller, Maddie Simens, Amanda Askell, Peter Welinder, Paul F. Christiano, Jan Leike, and Ryan Lowe.  Training language models to follow instructions with human feedback.  In _Proceedings of the Conference on Neural Information Processing Systems_ , 2022.
  * Qiao et al. (2023) Shuofei Qiao, Yixin Ou, Ningyu Zhang, Xiang Chen, Yunzhi Yao, Shumin Deng, Chuanqi Tan, Fei Huang, and Huajun Chen.  Reasoning with language model prompting: A survey.  In Anna Rogers, Jordan L. Boyd-Graber, and Naoaki Okazaki (eds.), _Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics_ , pp. 5368–5393. Association for Computational Linguistics, 2023.
  * Sanh et al. (2022) Victor Sanh, Albert Webson, Colin Raffel, Stephen H. Bach, Lintang Sutawika, Zaid Alyafeai, Antoine Chaffin, Arnaud Stiegler, Arun Raja, Manan Dey, M Saiful Bari, Canwen Xu, Urmish Thakker, Shanya Sharma Sharma, Eliza Szczechla, Taewoon Kim, Gunjan Chhablani, Nihal V. Nayak, Debajyoti Datta, Jonathan Chang, Mike Tian-Jian Jiang, Han Wang, Matteo Manica, Sheng Shen, Zheng Xin Yong, Harshit Pandey, Rachel Bawden, Thomas Wang, Trishala Neeraj, Jos Rozen, Abheesht Sharma, Andrea Santilli, Thibault Févry, Jason Alan Fries, Ryan Teehan, Teven Le Scao, Stella Biderman, Leo Gao, Thomas Wolf, and Alexander M. Rush.  Multitask prompted training enables zero-shot task generalization.  In _Proceedings of the 10th International Conference on Learning Representations_. OpenReview.net, 2022.
  * Shu et al. (2023) Yu Shu, Siwei Dong, Guangyao Chen, Wenhao Huang, Ruihua Zhang, Daochen Shi, Qiqi Xiang, and Yemin Shi.  Llasm: Large language and speech model.  _arXiv preprint arXiv:2308.15930_ , 2023.
  * Thoppilan et al. (2022) Romal Thoppilan, Daniel De Freitas, Jamie Hall, Noam Shazeer, Apoorv Kulshreshtha, Heng-Tze Cheng, Alicia Jin, Taylor Bos, Leslie Baker, Yu Du, YaGuang Li, Hongrae Lee, Huaixiu Steven Zheng, Amin Ghafouri, Marcelo Menegali, Yanping Huang, Maxim Krikun, Dmitry Lepikhin, James Qin, Dehao Chen, Yuanzhong Xu, Zhifeng Chen, Adam Roberts, Maarten Bosma, Yanqi Zhou, Chung-Ching Chang, Igor Krivokon, Will Rusch, Marc Pickett, Kathleen S. Meier-Hellstern, Meredith Ringel Morris, Tulsee Doshi, Renelito Delos Santos, Toju Duke, Johnny Soraker, Ben Zevenbergen, Vinodkumar Prabhakaran, Mark Diaz, Ben Hutchinson, Kristen Olson, Alejandra Molina, Erin Hoffman-John, Josh Lee, Lora Aroyo, Ravi Rajakumar, Alena Butryna, Matthew Lamm, Viktoriya Kuzmina, Joe Fenton, Aaron Cohen, Rachel Bernstein, Ray Kurzweil, Blaise Agüera y Arcas, Claire Cui, Marian Croak, Ed H. Chi, and Quoc Le.  Lamda: Language models for dialog applications.  _arXiv preprint arXiv:2201.08239_ , 2022.
  * Touvron et al. (2023a) Hugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martinet, Marie-Anne Lachaux, Timothée Lacroix, Baptiste Rozière, Naman Goyal, Eric Hambro, Faisal Azhar, Aurélien Rodriguez, Armand Joulin, Edouard Grave, and Guillaume Lample.  Llama: Open and efficient foundation language models.  _arXiv preprint arXiv:2302.13971_ , 2023a.
  * Touvron et al. (2023b) Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, Dan Bikel, Lukas Blecher, Cristian Canton-Ferrer, Moya Chen, Guillem Cucurull, David Esiobu, Jude Fernandes, Jeremy Fu, Wenyin Fu, Brian Fuller, Cynthia Gao, Vedanuj Goswami, Naman Goyal, Anthony Hartshorn, Saghar Hosseini, Rui Hou, Hakan Inan, Marcin Kardas, Viktor Kerkez, Madian Khabsa, Isabel Kloumann, Artem Korenev, Punit Singh Koura, Marie-Anne Lachaux, Thibaut Lavril, Jenya Lee, Diana Liskovich, Yinghai Lu, Yuning Mao, Xavier Martinet, Todor Mihaylov, Pushkar Mishra, Igor Molybog, Yixin Nie, Andrew Poulton, Jeremy Reizenstein, Rashi Rungta, Kalyan Saladi, Alan Schelten, Ruan Silva, Eric Michael Smith, Ranjan Subramanian, Xiaoqing Ellen Tan, Binh Tang, Ross Taylor, Adina Williams, Jian Xiang Kuan, Puxin Xu, Zheng Yan, Iliyan Zarov, Yuchen Zhang, Angela Fan, Melanie Kambadur, Sharan Narang, Aurélien Rodriguez, Robert Stojnic, Sergey Edunov, and Thomas Scialom.  Llama 2: Open foundation and fine-tuned chat models.  _arXiv preprint arXiv:2307.09288_ , 2023b.
  * Wang et al. (2023a) Haochun Wang, Chi Liu, Nuwa Xi, Zewen Qiang, Sendong Zhao, Bing Qin, and Ting Liu.  Huatuo: Tuning llama model with chinese medical knowledge.  _arXiv preprint arXiv:2304.06975_ , 2023a.
  * Wang et al. (2023b) Xuezhi Wang, Jason Wei, Dale Schuurmans, Quoc V. Le, Ed H. Chi, Sharan Narang, Aakanksha Chowdhery, and Denny Zhou.  Self-consistency improves chain of thought reasoning in language models.  In _Proceedings of the 11th International Conference on Learning Representations_. OpenReview.net, 2023b.
  * Wang et al. (2023c) Yizhong Wang, Yeganeh Kordi, Swaroop Mishra, Alisa Liu, Noah A. Smith, Daniel Khashabi, and Hannaneh Hajishirzi.  Self-instruct: Aligning language models with self-generated instructions.  In Anna Rogers, Jordan L. Boyd-Graber, and Naoaki Okazaki (eds.), _Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics_ , pp. 13484–13508. Association for Computational Linguistics, 2023c.
  * Wang et al. (2023d) Yue Wang, Hung Le, Akhilesh Deepak Gotmare, Nghi D. Q. Bui, Junnan Li, and Steven C. H. Hoi.  Codet5+: Open code large language models for code understanding and generation.  _arXiv preprint arXiv:2305.07922_ , 2023d.
  * Wei et al. (2022a) Jason Wei, Maarten Bosma, Vincent Y. Zhao, Kelvin Guu, Adams Wei Yu, Brian Lester, Nan Du, Andrew M. Dai, and Quoc V. Le.  Finetuned language models are zero-shot learners.  In _Proceedings of the 10th International Conference on Learning Representations_. OpenReview.net, 2022a.
  * Wei et al. (2022b) Jason Wei, Yi Tay, Rishi Bommasani, Colin Raffel, Barret Zoph, Sebastian Borgeaud, Dani Yogatama, Maarten Bosma, Denny Zhou, Donald Metzler, Ed H. Chi, Tatsunori Hashimoto, Oriol Vinyals, Percy Liang, Jeff Dean, and William Fedus.  Emergent abilities of large language models.  _Transactions on Machine Learning Research_ , 2022, 2022b.
  * Wei et al. (2022c) Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Brian Ichter, Fei Xia, Ed H. Chi, Quoc V. Le, and Denny Zhou.  Chain-of-thought prompting elicits reasoning in large language models.  In _Proceedings of the Conference on Neural Information Processing Systems_ , 2022c.
  * Wei et al. (2023) Tianwen Wei, Jian Luan, Wei Liu, Shuang Dong, and Bin Wang.  CMATH: can your language model pass chinese elementary school math test?  _arXiv preprint arXiv:2306.16636_ , 2023.
  * Wu et al. (2023) Shijie Wu, Ozan Irsoy, Steven Lu, Vadim Dabravolski, Mark Dredze, Sebastian Gehrmann, Prabhanjan Kambadur, David S. Rosenberg, and Gideon Mann.  Bloomberggpt: A large language model for finance.  _arXiv preprint arXiv:2303.17564_ , 2023.
  * Xu et al. (2023) Can Xu, Qingfeng Sun, Kai Zheng, Xiubo Geng, Pu Zhao, Jiazhan Feng, Chongyang Tao, and Daxin Jiang.  Wizardlm: Empowering large language models to follow complex instructions.  _arXiv preprint arXiv:2304.12244_ , 2023.
  * Yang et al. (2023a) Aiyuan Yang, Bin Xiao, Bingning Wang, Borong Zhang, Ce Bian, Chao Yin, Chenxu Lv, Da Pan, Dian Wang, Dong Yan, Fan Yang, Fei Deng, Feng Wang, Feng Liu, Guangwei Ai, Guosheng Dong, Haizhou Zhao, Hang Xu, Haoze Sun, Hongda Zhang, Hui Liu, Jiaming Ji, Jian Xie, Juntao Dai, Kun Fang, Lei Su, Liang Song, Lifeng Liu, Liyun Ru, Luyao Ma, Mang Wang, Mickel Liu, MingAn Lin, Nuolan Nie, Peidong Guo, Ruiyang Sun, Tao Zhang, Tianpeng Li, Tianyu Li, Wei Cheng, Weipeng Chen, Xiangrong Zeng, Xiaochuan Wang, Xiaoxi Chen, Xin Men, Xin Yu, Xuehai Pan, Yanjun Shen, Yiding Wang, Yiyu Li, Youxin Jiang, Yuchen Gao, Yupeng Zhang, Zenan Zhou, and Zhiying Wu.  Baichuan 2: Open large-scale language models.  _arXiv preprint arXiv:2309.10305_ , abs/2309.10305, 2023a.
  * Yang et al. (2023b) Hongyang Yang, Xiao-Yang Liu, and Christina Dan Wang.  Fingpt: Open-source financial large language models.  _arXiv preprint arXiv:2306.06031_ , 2023b.
  * Yu et al. (2023) Longhui Yu, Weisen Jiang, Han Shi, Jincheng Yu, Zhengying Liu, Yu Zhang, James T. Kwok, Zhenguo Li, Adrian Weller, and Weiyang Liu.  Metamath: Bootstrap your own mathematical questions for large language models.  _arXiv preprint arXiv:2309.12284_ , 2023.
  * Yue et al. (2023) Xiang Yue, Xingwei Qu, Ge Zhang, Yao Fu, Wenhao Huang, Huan Sun, Yu Su, and Wenhu Chen.  Mammoth: Building math generalist models through hybrid instruction tuning.  _arXiv preprint arXiv:2309.05653_ , 2023.
  * Zeng et al. (2023) Aohan Zeng, Xiao Liu, Zhengxiao Du, Zihan Wang, Hanyu Lai, Ming Ding, Zhuoyi Yang, Yifan Xu, Wendi Zheng, Xiao Xia, Weng Lam Tam, Zixuan Ma, Yufei Xue, Jidong Zhai, Wenguang Chen, Zhiyuan Liu, Peng Zhang, Yuxiao Dong, and Jie Tang.  GLM-130b: An open bilingual pre-trained model.  In _Proceedings of the 11th International Conference on Learning Representations_ , 2023.
  * Zhang et al. (2023a) Pan Zhang, Xiaoyi Dong, Bin Wang, Yuhang Cao, Chao Xu, Linke Ouyang, Zhiyuan Zhao, Shuangrui Ding, Songyang Zhang, Haodong Duan, Wenwei Zhang, Hang Yan, Xinyue Zhang, Wei Li, Jingwen Li, Kai Chen, Conghui He, Xingcheng Zhang, Yu Qiao, Dahua Lin, and Jiaqi Wang.  Internlm-xcomposer: A vision-language large model for advanced text-image comprehension and composition.  _arXiv preprint arXiv:2309.15112_ , 2023a.
  * Zhang et al. (2023b) Shengyu Zhang, Linfeng Dong, Xiaoya Li, Sen Zhang, Xiaofei Sun, Shuhe Wang, Jiwei Li, Runyi Hu, Tianwei Zhang, Fei Wu, and Guoyin Wang.  Instruction tuning for large language models: A survey.  _arXiv preprint arXiv:2308.10792_ , 2023b.
  * Zhao et al. (2023) Wayne Xin Zhao, Kun Zhou, Junyi Li, Tianyi Tang, Xiaolei Wang, Yupeng Hou, Yingqian Min, Beichen Zhang, Junjie Zhang, Zican Dong, Yifan Du, Chen Yang, Yushuo Chen, Zhipeng Chen, Jinhao Jiang, Ruiyang Ren, Yifan Li, Xinyu Tang, Zikang Liu, Peiyu Liu, Jian-Yun Nie, and Ji-Rong Wen.  A survey of large language models.  _arXiv preprint arXiv:2303.18223_ , 2023.
  * Zheng et al. (2023a) Chuanyang Zheng, Zhengying Liu, Enze Xie, Zhenguo Li, and Yu Li.  Progressive-hint prompting improves reasoning in large language models.  _arXiv preprint arXiv:2304.09797_ , 2023a.
  * Zheng et al. (2023b) Qinkai Zheng, Xiao Xia, Xu Zou, Yuxiao Dong, Shan Wang, Yufei Xue, Zihan Wang, Lei Shen, Andi Wang, Yang Li, Teng Su, Zhilin Yang, and Jie Tang.  Codegeex: A pre-trained model for code generation with multilingual evaluations on humaneval-x, 2023b.
  * Zhou et al. (2023) Yongchao Zhou, Andrei Ioan Muresanu, Ziwen Han, Keiran Paster, Silviu Pitis, Harris Chan, and Jimmy Ba.  Large language models are human-level prompt engineers.  In _Proceedings of the 11th International Conference on Learning Representations_. OpenReview.net, 2023.
  * Zhu et al. (2023) Deyao Zhu, Jun Chen, Xiaoqian Shen, Xiang Li, and Mohamed Elhoseiny.  Minigpt-4: Enhancing vision-language understanding with advanced large language models.  _arXiv preprint arXiv:2304.10592_ , 2023.
  * Zhu & Wang (2023) Wei Zhu and Xiaoling Wang.  Chatmed: A chinese medical large language model.  https://github.com/michael-wzhu/ChatMed, 2023.


# A short note on cospectral and integral chain graphs for Seidel matrix

Santanu Mandal<EMAIL_ADDRESS>
###### Abstract

In this brief communication, we investigate the cospectral as well integral
chain graphs for Seidel matrix, a key component to study the structural
properties of equiangular lines in space. We derive a formula that allows to
generate an infinite number of inequivalent chain graphs with identical
spectrum. In addition, we obtain a family of Seidel integral chain graphs.
This contrapositively answers a problem posed by Greaves ["Equiangular line
systems and switching classes containing regular graphs", Linear Algebra
Appl., (2018)] ("Does every Seidel matrix with precisely three distinct
rational eigenvalues contain a regular graph in its switching class?"). Our
observation is- "no".

AMS Classification:  05C50.
Keywords:  Chain graph, Seidel matrix, equiangular lines, switching,
cospectral, integral.

##  1 Introduction

Suppose $K_{n},C_{n}$ denote the complete graph and cycle on $n$ vertices
respectively. A graph with no induced subgraphs isomorphic to $C_{3},C_{5}$
and $2K_{2}$ is called a chain graph. Chain graph has several equivalent
definitions. In this study, we consider the definition based on binary string,
that will be relevant to this work. The underlying matrix is the Seidel matrix
$S$.
An $n$ vertex chain graph can be generated by the binary string
$b=0^{s_{1}}1^{t_{1}}0^{s_{2}}\ldots 0^{s_{k}}1^{t_{k}}$, where
$s_{i},t_{i}\geq 1$ and $\sum s_{i}+\sum t_{i}=n$ (see [8]). Note that a chain
graph is a bipartite graph, and it is complete bipartite if and only if $k=1$.
Let $G=(V,E)$ be a simple, connected and finite graph with vertex set
$V=\\{v_{1},v_{2},\ldots,v_{n}\\}$. Let $A$ be the $(0,1)$-adjacency matrix of
$G$. Then the Seidel matrix (studied in [1, 8]) is a square matrix of order
$n$ defined by

$S=J-I-2A,$

where $J$ is all $1$ matrix and $I$ is the identity matrix. In other words, if
$s_{ij}$ is the $(i,j)$-th entry of $S$, then

$s_{ij}=\begin{cases}-1&\text{if }v_{i}\sim v_{j},\\\ 1&\text{if }v_{i}\nsim
v_{j},\ i\neq j,\\\ 0&\text{if }i=j.\end{cases}$

###  1.1 Equiangular line

A system of lines $\mathcal{L}$ in the Euclidean $d-$dimensional space
$\mathbb{R}^{d}$ is said to be equiangular if all the lines meet at a point
and have pairwise equal angle. The constant angle is called the common angle
of the system. There is a closed correspondence between the equiangular line
and $(-1,0,1)-$Seidel matrix. Seidel and Lint [6] first notice this important
relationship around $1966$. Later Greaves et al. [2, 3, 4] develope several
significant results in the theory of equiangular lines in connection with
Seidel matrix. In fact, Seidel matrix plays a key role to study the structure
of equiangular lines in space geometry. Seidel matrix can be viewed as a Gram
matrix corresponding to an equiangular line system of $n>d$ lines in
$\mathbb{R}^{d}$, where the pairwise inner product between any two distinct
lines is $\pm\frac{1}{\lambda_{min}}$ (see [6]), where $\lambda_{min}$ is the
smallest eigenvalue of $S$. Suppose $S$ is the Seidel matrix of a graph with
$n$ vertices and $\lambda_{min}$ is the least eigenvalue with multiplicity
$(n-d)$ for some natural number $d$. Then $\mathcal{L}$ contains at most $n$
equiangular lines in the Euclidean space $\mathbb{R}^{d}$. The common angle is
$\frac{1}{\lambda_{min}}$. The study of Seidel matrices is motivated by this
correspondence, which also emphasises the significance of their smallest
eigenvalue. We believe that this work can be of interest to the readers for
its tight connection with equiangular lines in the Euclidean spaces (see [2,
3, 4, 6, 10] for more details).

###  1.2 Switching

Suppose $G=(V,E)$ is a graph and $V^{{}^{\prime}}\subseteq V$. Then the
switching graph of $G$ on $V^{{}^{\prime}}$ is the graph $G^{{}^{\prime}}$
arises from $G$ by changing all edges between $V^{{}^{\prime}}$ and
$V\setminus V^{{}^{\prime}}$ to non-edges, and all the non-edges to edges as
well. This operation is called the switching on the subset $V^{{}^{\prime}}$.
The adjacency property among the vertices in $V$ (or in $V^{{}^{\prime}}$) is
preserved when switching is applied. That means we leave the edges and non-
edges within $V^{{}^{\prime}}$ (or $V\setminus V^{{}^{\prime}}$) unaltered.
The collection of all graphs that can be obtained from $G$ by switching on
every possible subset of $V$ is called the switching class of $G$. Two graphs
$G$ and $H$ are called the switching equivalent (or simply equivalent) if they
belong to the same switching class. Otherwise, we call them simply
inequivalent. If two graphs are equivalent (res. inequivalent) then their
Seidel matrices are said to be equivalent (res. inequivalent). Equivalent
Seidel matrices are cospectral, but starting from $n=8$ examples of
inequivalent cospectral Seidel matrices (of order $n$) appear. That means
there does not exist any cospectral graphs (which are not switching
equivalent) of order $n\leq 7$. Szöllösi et al. [10] tabulated the number of
inequivalent cospectral Seidel matrices of order $n$ where $n\leq 13$.
However, here we construct an infinite family of inequivalent cospectral chain
graphs of order $n$ where $n\geq 14$. Greaves [2] investigated the case- when
a Seidel matrix having precisely three distinct eigenvalues has a regular
graph in its switching class. He noted that, excluding one (this contains
irrational eigenvalues), every Seidel matrix of order at most $12$ with
precisely three distinct eigenvalues has a regular graph in its switching
class. And, motivated from this observation, he left a question:
Question 1.[2] Does every Seidel matrix with precisely three distinct rational
eigenvalues contain a regular graph in its switching class?
We are able to answer this question contrapositively by finding one counter
example (see Corollary 3.7). Szöllösi and Östergárd [10] pointed out an
infinite family of Seidel matrices (irrespective of the nature of the
eigenvalues) that do not have a regular graph in their switching classes.
Recall that a $\\{P_{4},C_{4},2K_{2}\\}-$ free graph is called a threshold
graph. Mandal and Mehatari [9] proved that - no threshold graphs can have
three distinct Seidel eigenvalues. In light of this conclusion, a
straighforward theorem is given below.

######  Theorem 1.1.

No threshold graphs with three distinct Seidel eigenvalues contain a regular
graph in its switching class.

###  1.3 Quotient Matrix

Let us consider a chain graph $G$ with the binary string
$b=0^{s_{1}}1^{t_{1}}0^{s_{2}}\ldots 0^{s_{k}}1^{t_{k}}$. Then the Seidel
matrix $S$ of $G$ is a square matrix of size $n$, given by

$S=\begin{bmatrix}(J-I)_{s_{1}}&-J_{s_{1}\times t_{1}}&J_{s_{1}\times
s_{2}}&-J_{s_{1}\times t_{2}}&J_{s_{1}\times s_{3}}&-J_{s_{1}\times
t_{3}}&\ldots&-J_{s_{1}\times t_{k}}\\\\[2.84526pt] -J_{t_{1}\times
s_{1}}&(J-I)_{t_{1}}&J_{t_{1}\times s_{2}}&J_{t_{1}\times
t_{2}}&J_{t_{1}\times s_{3}}&J_{t_{1}\times t_{3}}&\ldots&J_{t_{1}\times
t_{k}}\\\\[2.84526pt] J_{s_{2}\times s_{1}}&J_{s_{2}\times
t_{1}}&(J-I)_{s_{2}}&-J_{s_{2}\times t_{2}}&J_{s_{2}\times
s_{3}}&-J_{s_{2}\times t_{3}}&\ldots&-J_{s_{2}\times t_{k}}\\\\[2.84526pt]
-J_{t_{2}\times s_{1}}&J_{t_{2}\times t_{1}}&-J_{t_{2}\times
s_{2}}&(J-I)_{t_{2}}&J_{t_{2}\times s_{3}}&J_{t_{2}\times
t_{3}}&\ldots&J_{t_{2}\times t_{k}}\\\\[2.84526pt] &&&&&&\ddots\\\\[2.84526pt]
J_{s_{k}\times s_{1}}&J_{s_{k}\times t_{1}}&J_{s_{k}\times
s_{2}}&J_{s_{k}\times t_{2}}&J_{s_{k}\times s_{3}}&J_{s_{k}\times
t_{3}}&\ldots&-J_{s_{k}\times t_{k}}\\\\[2.84526pt] -J_{t_{k}\times
s_{1}}&J_{t_{k}\times t_{1}}&-J_{t_{k}\times s_{2}}&J_{t_{k}\times
t_{2}}&-J_{t_{k}\times s_{3}}&J_{t_{k}\times
t_{3}}&\ldots&(J-I)_{t_{k}}\end{bmatrix},$

where $J_{m\times n}$ is all 1 block matrix of size $m\times n$ and the
diagonal blocks of $S$ are the square matrices of size $s_{1}\times
s_{1},~{}t_{1}\times t_{1},~{}s_{2}\times s_{2},~{}t_{2}\times
t_{2},~{}\ldots,~{}t_{k}\times t_{k}$.
We now construct an equitable partition in the following way. For the
representation $b=0^{s_{1}}1^{t_{1}}0^{s_{2}}\ldots 0^{s_{k}}1^{t_{k}}$ of
$G$, let $V_{s_{1}}$ denote the set of vertices representing first $s_{1}$
added vertices, $V_{t_{1}}$ denote the set of vertices representing next
$t_{1}$ added vertices, $V_{s_{2}}$ denote the set of vertices representing
next $s_{2}$ added vertices and finally $V_{t_{k}}$ denote the set of vertices
representing last $t_{k}$ added vertices. Now we consider the vertex partition
of $\pi=\\{C_{1},C_{2},C_{3},\ldots,C_{2k}\\}$, where $C_{i}=V_{s_{j}}=s_{j}$,
if $i=2j-1$ $(j=1,\,2,\ldots,\,k)$, and $C_{i}=V_{t_{j}}=t_{j}$, if $i=2j$
$(j=1,\,2,\ldots,\,k)$. Then $\pi$ is an equitable partition of $G$ of size
$2k$. Throughout this paper we consider this equitable partition only, and by
writing “$G$ is a chain graph of order $n$ with $|\pi|=2k$” we mean that $G$
is a chain graph of order $n$ with the binary string
$b=0^{s_{1}}1^{t_{1}}0^{s_{2}}\ldots 0^{s_{k}}1^{t_{k}}$. Let $Q_{\pi}$ be the
corresponding quotient matrix. Then $Q_{S}$ is a square matrix of size $2k$,
given by

$Q_{\pi}=\begin{bmatrix}(s_{1}-1)&-t_{1}&s_{2}&-t_{2}&s_{3}&-t_{3}&\ldots&-t_{k}\\\\[2.84526pt]
-s_{1}&(t_{1}-1)&s_{2}&t_{2}&s_{3}&t_{3}&\ldots&t_{k}\\\\[2.84526pt]
s_{1}&t_{1}&(s_{2}-1)&-t_{2}&s_{3}&-t_{3}&\ldots&-t_{k}\\\\[2.84526pt]
-s_{1}&t_{1}&-s_{2}&(t_{2}-1)&s_{3}&t_{3}&\ldots&t_{k}\\\\[2.84526pt]
s_{1}&t_{1}&s_{2}&t_{2}&(s_{3}-1)&-t_{3}&\ldots&-t_{k}\\\\[2.84526pt]
-s_{1}&t_{1}&-s_{2}&t_{2}&-s_{3}&(t_{3}-1)&\ldots&t_{k}\\\\[2.84526pt]
&&&&&&\ddots\\\\[2.84526pt]
s_{1}&t_{1}&s_{2}&t_{2}&s_{3}&t_{3}&\ldots&-t_{k}\\\\[2.84526pt]
-s_{1}&t_{1}&-s_{2}&t_{2}&-s_{3}&t_{3}&\ldots&(t_{k}-1)\end{bmatrix}.$

The following results are helpful in this study.

######  Theorem 1.2.

[8] Let $G$ be a chain graph of order $n$ with $|\pi|=2k$. Then

$n_{-1}(S)=n-2k+1,$

where $n_{-1}(S)$ is the multiplicity of the eigenvalue $-1$ of $S$.

######  Theorem 1.3.

[8] An eigenvalue of $Q_{\pi}$ can appear at most twice.

##  2 Cospectral chain graphs

Construction of cospectral graphs with respect to Seidel matrix is more
difficult than any other graph matrices (like adjacency, Laplacian, distance
etc). Because for the most part, the idea of switching-equivalent prevents the
occurrence of such graphs in the Seidel matrix scenario. The creation of
cospectral graphs with respect to Seidel matrix is a challenging work because
of this. We call two graphs Seidel cospectral if they are cospectral on Seidel
matrix. Berman et al. [1] construct a family of complete tripartite graphs
which are Seidel cospectral. Here we produce a family of Seidel cospectral
chain graphs. Let us first prove the following lemma which confirms the
inequivalency of two chain graphs.

######  Lemma 2.1.

Two chain graphs with the binary strings $B_{1}=01^{a_{1}}0^{a_{1}}1^{b_{1}}$
and $B_{2}=01^{a_{2}}0^{a_{2}}1^{b_{2}}$ are NOT switching equivalent, where
$a_{1}\neq a_{2}$, but $2a_{1}+b_{1}=2a_{2}+b_{2}$.

###### Proof.

Let $V_{i}$ be the independent sets in $B_{1}$ such that
$|V_{1}|=1,|V_{2}|=|V_{3}|=a_{1},|V_{4}|=b_{1}$. Also let $W_{i}$ be the
independent sets in $B_{2}$ such that
$|W_{1}|=1,|W_{2}|=|W_{3}|=a_{2},|W_{4}|=b_{2}$. Suppose there exists
$U\in\cup_{i=1}^{4}V_{i}$ such that switching $B_{1}$ with respect to $U$
produces the chain graph $G^{{}^{\prime}}=B_{2}$. Now there arise following
cases.

Case 1. Let $x\in U$. If $x$ belongs to either of the vertex sets $V_{1}$,
$V_{2}$, or $V_{3}$, then switching $B_{1}$ with respect to $U$ produces no
chain graph.

Case 2. Let $U=\\{x_{i}\in V_{4}$ : $i=1,2,3,\cdots,q$}. Then switching
$B_{1}$ with respect to $U$ produces the chain graph switching-equivalent to
$G^{{}^{\prime}}=0^{q+1}1^{a_{1}}0^{a_{1}}1^{b_{1}-q}$.

Case 3. Let $U=\\{x,y:x\in V_{1},y\in V_{4}\\}$. Then switching $B_{1}$ with
respect to $U$ produces the chain graph switching-equivalent to
$G^{{}^{\prime}}=01^{a_{1}}0^{a_{1}}1^{b_{1}}.$

Case 4. Let $U=\\{V_{i}\cup V_{j}\\}$. Now if $(i,j)\in\\{(1,4),(2,3)\\}$,
then switching $B_{1}$ with respect to $U$ produces the chain graph switching-
equivalent to $G^{{}^{\prime}}=01^{a_{1}}0^{a_{1}}1^{b_{1}}.$

Case 5. Let $U=\\{V_{i}\cup V_{j}\\}$. Now if
$(i,j)\in\\{(3,4),(1,3),(1,2),(3,3)\\}$, then switching $B_{1}$ with respect
to $U$ produces no chain graph.
Therefore from the above cases we conclude that any switching on $B_{1}$ does
not produce $B_{2}$ if $a_{1}\neq a_{2}$. This completes the proof of the
lemma. ∎

Motivated from this lemma we construct a family of Seidel cospectral chain
graphs (inequivalent) as follows.

######  Theorem 2.2.

The chain graphs $b_{1}=01^{m}0^{m}1^{2m+r}$ and $b_{2}=01^{2m}0^{2m}1^{r}$
are Seidel cospectral if there exists an odd positive integer $r$ such that

$m=\frac{3}{2}(r+1)$

.

###### Proof.

Note that the given chain graphs are not switching-equivalent as demonstrated
in the previous lemma. Their eigenvalues are

Graph | Eigenvalues
---|---
$01^{m}0^{m}1^{2m+r}$ | $-1^{4m+4r-2},~{}2m-1,~{}\frac{1}{2}(2m+r-1\pm\sqrt{20m^{2}+12mr+12m+r^{2}+2r+1})$
$01^{2m}0^{2m}1^{r}$ | $-1^{4m+4r-2},~{}4m-1,~{}\frac{1}{2}(r-1\pm\sqrt{16mr+16m+r^{2}+2r+1})$

There arise three cases.
Case 1. $4m-1=\frac{1}{2}(2m+r-1+\sqrt{20m^{2}+12mr+12m+r^{2}+2r+1})$.
Case 2. $2m-1=\frac{1}{2}(r-1+\sqrt{16mr+16m+r^{2}+2r+1})$.
Case 3.
$\frac{1}{2}(2m+r-1-\sqrt{20m^{2}+12mr+12m+r^{2}+2r+1})=\frac{1}{2}(r-1-\sqrt{16mr+16m+r^{2}+2r+1})$.
On simplification of the above three cases, we obtain the required result. ∎

######  Remark 2.3.

This classifies the chain graphs with exactly four distinct Seidel
eigenvalues.

######  Corollary 2.4.

In the above theorem, when cospectrality occurs then the spectral radius is
$n-(r+2)$.

######  Corollary 2.5.

In Theorem 2.2, when cospectrality occurs then the smallest eigenvalue is
simple and given by the formula $\lambda_{min}=-(2m-r)$.

As a quick application of this result, here one example is considered to find
the number of equiangular lines in the $d-$ dimensional Euclidean space
$\mathbb{R}^{d}$ for some positive integer $d$.

######  Example 2.6.

In the Theorem 2.2, when cospectrality occurs, then the least eigenvalue is
$-(2m-r)$ which occurs once. Note that the order of the Seidel matrix is
$4m+r+1$. The corresponding $4m+r+1$ lines form the largest known equiangular
line system in $\mathbb{R}^{4m+r}$. And, the common angle is $\frac{1}{2m-r}$.

We now give a list of $10$ pair of Seidel cospectral chain graphs in the
following table (see Table 1).

No. of vertices | Chain graph$(01^{m}0^{m}1^{2m+r})$ | chain graph$(01^{2m}0^{2m}1^{r})$ | Eigenvalues
---|---|---|---
$14$ | $01^{3}0^{3}1^{7}$ | $01^{6}0^{6}1$ | $-1^{11},\pm 5,11$
$28$ | $01^{6}0^{6}1^{15}$ | $01^{12}0^{12}1^{3}$ | $-1^{25},-9,11,23$
$42$ | $01^{9}0^{9}1^{23}$ | $01^{18}0^{18}1^{5}$ | $-1^{39},-13,17,35$
$56$ | $01^{12}0^{12}1^{31}$ | $01^{24}0^{24}1^{7}$ | $-1^{53},-17,23,47$
$70$ | $01^{15}0^{15}1^{39}$ | $01^{30}0^{30}1^{9}$ | $-1^{67},-21,29,59$
$84$ | $01^{18}0^{18}1^{47}$ | $01^{36}0^{36}1^{11}$ | $-1^{81},-25,35,71$
$98$ | $01^{21}0^{21}1^{55}$ | $01^{42}0^{42}1^{13}$ | $-1^{95},-29,41,83$
$112$ | $01^{24}0^{24}1^{63}$ | $01^{48}0^{48}1^{15}$ | $-1^{109},-33,47,95$
$126$ | $01^{27}0^{27}1^{71}$ | $01^{54}0^{54}1^{17}$ | $-1^{123},-37,53,107$
$140$ | $01^{30}0^{30}1^{79}$ | $01^{60}0^{60}1^{19}$ | $-1^{137},-41,59,119$
Table 1: Cospectral chain graphs

##  3 Integral chain graphs

Which graphs have integral spectra? This was first raised by Harary et el. [5]
in $1974$. After that the problem of finding integral graphs become more
popular. Integral graphs are actually quite uncommon. We call a graph Seidel
integral if the Seidel matrix consists entirely of integer eigenvalue. In this
section we try to obtain few Seidel integral chain graphs. Zhao et al. [7]
produced Seidel integral complete multipartite graphs. They obtained a
necessary and sufficient condition for a complete tripartite graph $K_{p,q,r}$
to be Seidel integral. Whereas Wang et al. [11] provided a list of Seidel
integral complete r-partite graphs. Suppose $G$ is a chain graph on $n$
vertices with the binary string $b=0^{s_{1}}1^{t_{1}}0^{s_{2}}\ldots
0^{s_{k}}1^{t_{k}}$. Here we obtain a family of Seidel integral chain graphs
for $k=2$.

######  Theorem 3.1.

The chain graph with binary string $0^{s}1^{2s}0^{2s}1^{s}$ is Seidel
integral.

###### Proof.

The Seidel eigenvalues of the given chain graph are

$-1^{6s-3},-2s-1,4s-1,4s-1,$

where exponents denote the frequency of occurrence of eigenvalue. ∎

######  Remark 3.2.

Theorem 3.1 classifies the chain graphs with exactly three distinct (rational)
Seidel eigenvalues such that its switching class contains a regular graph of
valency $3s$. This can be viewed as an example in support of a question posed
by Greaves (Question A, [2]) affirmatively.

######  Corollary 3.3.

For the chain graph $0^{s}1^{2s}0^{2s}1^{s}$, $\lambda_{min}=-(2s+1)$ with
multiplicity $2$.

######  Theorem 3.4.

The $n$ vertex chain graph with binary string $01^{m}0^{m}1^{n-2m-1}$ is
Seidel integral if $(n,m)$ takes any one of the following values

$(n,m)\in\begin{cases}\\{(3r,r)\\}&\text{for }r\geq 2,\\\
\\{(13r,6r),(13r,2r)\\}&\text{for }r\geq 2,\\\
\\{(2r^{2}+2r+2,r^{2}+r)\\}&\text{for }r\geq 1,\\\
\\{(4r^{2}-2r+1,r)\\}&\text{for }r\geq 3,\\\
\\{(4r^{2}+4r+4,2r^{2}+2r)\\}&\text{for }r\geq 1,\end{cases}$

where $r$ is a positive integer.

###### Proof.

The Seidel eigenvalues of the given chain graph are

$-1^{n-3},~{}2m-1,~{}-(m+1)+\frac{1}{2}\Big{[}n\pm\sqrt{(n-2m)(n+6m)}\Big{]},$

where exponents denote the frequency of occurrence of eigenvalue. Note that
$(n-2m)(n+6m)$ is even (res. odd) if $n$ is even (res. odd). Therefore the
eigenvalues are integral if the term $(n-2m)(n+6m)$ is a perfect square. Now,
$(n-2m)(n+6m)=(n+2m)^{2}-(4m)^{2}=x^{2}-y^{2}$, (say), where $x=n+2m$, $y=4m$.
Since $n>2m$, so $x>y$. Let $x=y+h$. Therefore

$n=h+2m$ (1)

Now, $x^{2}-y^{2}=(y+h)^{2}-y^{2}=h(2y+h)$. It is easy to verify that

$h^{2}<h(2y+h)<4(h+2m)^{2},$

which means that $(n-2m)(n+6m)$ can takes the perfect square value lie in the
open interval $(h^{2},4(h+2m)^{2})$. The possible perfect square which lie in
the open interval $(h^{2},4(h+2m)^{2})$ are
$(h+m)^{2},(h+2m)^{2},(h+3m)^{2},(h+2r)^{2},\\{h+m+(2p-1)\\}^{2}~{}for~{}p\geq
3,(h+i)^{2}~{}for~{}i=1,2,3,4,8$. We now consider several cases to obtain the
values of $m,n$ such that the expression within the square root of the
eigenvalues become perfect square.
Case 1. $h(2y+h)=(h+m)^{2}$. Simplifying and using equation no. (1) we obtain

$(n,m)=(13r,6r),$

where $r$ is an integer $\geq 2$.
Case 2. $h(2y+h)=(h+2m)^{2}$. Simplifying and using equation no.(1) we obtain

$(n,m)=(3r,r),$

where $r$ is an integer $\geq 2$.
Case 3. $h(2y+h)=(h+3m)^{2}$. Simplifying and using equation no. (1) we obtain

$(n,m)=(13r,2r),$

where $r$ is an integer $\geq 2$.
Case 4. $h(2y+h)=(h+i)^{2}~{}for~{}i=1,2,3,4,8$. We see that for $i=1,2,3$ the
value of $h$ becomes fractional, an absurd result. So $i=4,8$ are the only
acceptable case. Simplifying and using equation no. (1) we obtain

$(n,m)\in\\{(6,2),(14,6),(12,4)\\}.$

Case 5. $h(2y+h)=(h+2r)^{2}$. This gives

$h=\frac{4r^{2}}{8m-4r}$ (2)

which should be an integer value. Now the divisors of $4r^{2}$ are
$2,4,r,r^{2},2r,2r^{2},4r,4r^{2}$. So we take the following subcases.
Subcase a. $8m-4r=4$. Simplifying and using equations no. (1), (2) we obtain

$(n,m)=(4r^{2}-2r+1,r),$

where $r$ is an integer $\geq 3$.
Subcase b. $8m-4r=4r$. Simplifying and using equations no. (1), (2) we obtain

$(n,m)=(3r,r),$

where $r$ is an integer $\geq 2$.
Subcase c. $8m-4r=r^{2}$. Simplifying and using equations no. (1), (2) we
obtain

$(n,m)=(4r^{2}+4r+4,2r^{2}+2r),$

where $r$ is an integer.
Subcase d. $8m-4r=2r^{2}$. Simplifying and using equations no. (1), (2) we
obtain

$(n,m)=(2r^{2}+2r+2,r^{2}+r),$

where $r$ is a positive integer.
We discard all other cases where absurd result found. This completes the proof
of the theorem. ∎

######  Corollary 3.5.

For the chain graph $01^{m}0^{m}1^{n-2m-1}$, the minimum eigenvalue is given
by

$\lambda_{min}=-(m+1)+\frac{1}{2}\Big{[}n-\sqrt{(n-2m)(n+6m)}\Big{]}.$

######  Remark 3.6.

When cospectrality occurs then Theorem 2.2 produces an infinite family of
Seidel integral chain graph. Some examples of Seidel integral chain graphs are
depicted in Table 1 and Table 2, where exponents denote the repetition of an
eigenvalue.

Chain graph$(0^{s}1^{2s}0^{2s}1^{s})$ | Eigenvalues | chain graph $(01^{m}0^{m}1^{n-2m-1})$ | Eigenvalues
---|---|---|---
$01^{2}0^{2}1$ | $-1^{3},-3,3^{2}$ | $01^{3}0^{3}1^{2}$ | $-1^{6},-4,5^{2}$
$0^{2}1^{4}0^{4}1^{2}$ | $-1^{9},-5,7^{2}$ | $01^{3}0^{3}1^{24}$ | $-1^{28},-6,5,29$
$0^{3}1^{6}0^{6}1^{3}$ | $-1^{15},-7,11^{2}$ | $01^{4}0^{4}1^{48}$ | $-1^{54},-8,7,55$
$0^{4}1^{8}0^{8}1^{4}$ | $-1^{21},-9,15^{2}$ | $01^{6}0^{6}1$ | $-1^{11},\pm 5,11$
$0^{5}1^{10}0^{10}1^{5}$ | $-1^{27},-11,19^{2}$ | $01^{5}0^{5}1^{80}$ | $-1^{88},-10,9,89$
$0^{6}1^{12}0^{12}1^{6}$ | $-1^{33},-13,23^{2}$ | $01^{4}0^{4}1^{3}$ | $-1^{9},-5,7^{2}$
$0^{7}1^{14}0^{14}1^{7}$ | $-1^{39},-15,27^{2}$ | $01^{12}0^{12}1$ | $-1^{23},\pm 7,23$
$0^{8}1^{16}0^{16}1^{8}$ | $-1^{45},-17,31^{2}$ | $01^{4}0^{4}1^{17}$ | $-1^{23},\pm 7,23$
$0^{9}1^{18}0^{18}1^{9}$ | $-1^{51},-19,35^{2}$ | $01^{5}0^{5}1^{4}$ | $-1^{12},-6,9^{2}$
$0^{10}1^{20}0^{20}1^{10}$ | $-1^{57},-21,39^{2}$ | $01^{6}0^{6}1^{26}$ | $-1^{36},-10,11,35$
Table 2: Integral chain graphs

######  Corollary 3.7.

From Table 2, we see that the chain graph $G$ on $15$ vertices with binary
string $01^{5}0^{5}1^{4}$ has three distinct rational eigenvalues:
$-1^{12},-6,9^{2}$ (exponents denote the frequency of occurrence) such that
its switching class does not contain a regular graph. We verify this result
manually by switching $G$ with respect to all possible subset of $15$
vertices. We notice that no switching produces a regular graph but rather a
bi-regular graph (of valencies $7~{}\&~{}8$). This occurs when we apply
switching with respect to $7$ vertices ($2$ from $1^{5}$, $3$ from $0^{5}$ and
$2$ from $1^{4}$). Therefore, the existence of such a graph negates the
Greave’s question.

##  4 Acknowledgments

The research of Santanu Mandal is supported by the University Grants
Commission of India under the beneficiary code BININ01569755.

##  5 Statements and Declarations

Competing Interests: The author made no mention of any potential conflicts of
interest.

## References

  * [1] A. Berman, N. S. Monderer, R. Singh, X. D. Zhang, Complete multipartite graphs that are determined , up to switching, by their Seidel spectrum, Linear Algebra Appl., 564 (2019), 58–71.
  * [2] G. R. W. Graves, Equiangular line systems and switching classes containing regular graphs, Linear Algebra Appl., 536 (2018) 31-51.
  * [3] G. R. W. Graves, J. H. Koolen, A. Munemasa, F. Szöllösi, Equiangular lines in Euclidean spaces J. Combin. Theory-Series A, 138 (2016) 208-235.
  * [4] G. R. W. Graves, J. Syatriadi, P. Yatsyna, Equiangular lines in low dimensional euclidean spaces, Combinatorica, 41 (6) (2021) 839-872.
  * [5] F. Harary, A. J. Schwenk, Which graphs have integral spectra?, Graphs Combin., Springer-Verlag, Berlin (1974), 45-51.
  * [6] J. H. Van Lint, J. J. Seidel, Equilateral point sets in elliptic geometry, Indag. Math., 28 (1966) 335-348.
  * [7] S. LV, L. Wei, H. Zhao, On the Seidel integral complete multipartite graphs, Acta Math. Appl., 28, 4 (2012) 705-710.
  * [8] S. Mandal, R. Mehatari and K C Das, On the spectrum and energy of Seidel matrix for chain graphs, (2022), Preprint: available at- arXiv:2205.00310.
  * [9] S. Mandal, R. Mehatari, On the Seidel spectrum of threshold graphs, Indian J. Pure Appl. Math., (2023), DOI- https://doi.org/10.1007/s13226-023-00436-9
  * [10] F. Szöllösi, P. R. J. Östergárd, Enumeration of Seidel matrices, European J. Combin., 69 (2018), 169–184.
  * [11] L. Wang, G. Zhao, K. Li, Seidel integral complete r-partite graphs, Graphs Combin., (2012), DOI: 10.1007/s00373-012-1276-6.