# Article ID: MTJPAM-D-21-00024

## Title: Power Exponential Mean Labeling of Graphs

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-21-00024; Volume 3 / Issue 2 / Year 2021, Pages 70-79

Document Type: Research Paper

Author(s): Kurugal Munikempanna Nagaraja a , Sampathkumar Ramachandraiah b , Venkataramana Bathahalli Siddappa c

bDepartment of Mathematics, R N S Institute of Technology, Uttarahalli – Kengeri Main Road, R R Nagar post, Bengaluru, Karnataka, India

cDepartment of Mathematics, K S Institute of Technology, Kannakapura Main Road, Bengaluru, Karnataka, India

Received: 18 March 2021, Accepted: 14 May 2021, Published: 12 June 2021.

Corresponding Author: Kurugal Munikempanna Nagaraja (Email address: nagkmn@gmail.com)

Abstract

A $(p, q)$  graph $G$  is said to be a power exponential mean graph if there exist a one to one correspondence $f:V\rightarrow\{1,2,3,\ldots,p\}$  such that induced function $f^*:E(G)\rightarrow N$  given by $f^* (uv)=\left\lceil{\left({f(u) }^{f(u)}{f(v)}^{f(v)}\right)}^{\frac{1}{f(u)+f(v)}}\right\rceil \qquad \text{or} \qquad f^* (uv)=\left\lfloor{\left({f(u) }^{f(u)}{f(v)}^{f(v)}\right)}^{\frac{1}{f(u)+f(v)}}\right\rfloor$

for every $uv\in E(G)$  are all distinct. In this paper the power exponential mean labeling of graphs such as path, cycle, $K_1+C_n$  for $n$  is odd and even, square graph, umbrella $U(m,n)$, duplicating each vertex by an edge in path $P_n$, comb, $C_m\bigodot \overline{ K_1}$  and $C_n\bigodot \overline{ K_2}$  are discussed.

Keywords: Graph, power exponential mean, path, cycle, square graph, umbrella, comb

References:
1. A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad, Products of finite groups, de Gruyter Expositions in Mathematics, 53 Walter de Gruyter GmbH & Co. KG, Berlin, 2010.
2. P. S. Bullen, Handbook of means and their inequalities, Kluwer Acad. Publ., Dordrecht, 2003.
3. Gy. Elekes, A note on a problem of Erdös on right angles, Discrete Math. 309 (16), 5253–5254, 2009.
4. J. A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of combinatorics 2013.
5. G. Glauberman, A p-group with no normal large abelian subgroup, Character theory of finite groups, 61–65, Contemp. Math., 524, Amer. Math. Soc., Providence, RI, 2010.
6. F. Harary, Graph theory, Narosa publishing House, New Delhi, 1988.
7. C. Jayasekaran and C. David Raj, Harmonic mean labeling of disconnected graphs, Journal of Discrete Mathematical Sciences and Cryptography 19 (1), 1–12, 2016.
8. K. M. Nagaraja, V. Lokesha and S. Padmanabhan A simple proof on strengthening and extension of inequalities, Adv. Stud. Contemp. Math. 17 (1), 97–103, 2008.
9. K. M. Nagaraja, R. Sampath Kumar, B. S. Venkataramana and A. Harish, Ky Fan type inequality chain involving power exponential mean and its invariant, GIS Science Journal 8 (1), 105–115, 2021.
10. R. Sampath Kumar, G. Narasimhan and R. Chandrasekhar, Contra harmonic mean labeling of graph, Mapana J Sci. 12 (3), 23–29, 2013.
11. R. Sampath Kumar, G. Narasimhan and K. M. Nagaraja, Heron mean labeling of graph, International Journal of Recent Scientific Research, 8 (9), 19808–19811, 2017.
12. R. Sampath Kumar and K. M. Nagaraja, Centroidal mean labeling of graphs, International Journal of Research in Advent Technology 7 (1), 183–189, 2019.
13. S. S. Sandhya, S. Somasundaram and R. Ponraj, Some more results on harmonic mean graphs, Journal of Mathematics Research 4 (1), 21–29, 2012.
14. S. Somasundaram and R. Ponraj, Mean labeling of graphs, National Academy of Science Letters 26 (7), 210–213, 2003.
15. S. Somasundaram and P. Vidhyarani, Geometric mean labeling of graphs, Bulletin of Pure and Applied Sciences 30 (2), 153–160, 2011.
{"ft_lang_label":"__label__en","ft_lang_prob":0.7205316,"math_prob":0.8572942,"size":3382,"snap":"2023-40-2023-50","text_gpt3_token_len":996,"char_repetition_ratio":0.10390764,"word_repetition_ratio":0.019011406,"special_character_ratio":0.29183915,"punctuation_ratio":0.26933703,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98305315,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],"im_url_duplicate_count":[null,8,null,8,null,8,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-27T19:05:56Z\",\"WARC-Record-ID\":\"<urn:uuid:6b930add-a482-48cc-9770-bd250d033eeb>\",\"Content-Length\":\"71234\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b77dddee-8b9e-40ba-b7c4-07f47452a1d6>\",\"WARC-Concurrent-To\":\"<urn:uuid:d9a5d5c0-ec01-44fe-9b84-79bc9cca0fe8>\",\"WARC-IP-Address\":\"192.0.78.129\",\"WARC-Target-URI\":\"https://mtjpamjournal.com/papers/article_id_mtjpam-d-21-00024/\",\"WARC-Payload-Digest\":\"sha1:WAFYXZYHGGB2GWZ2SCVRXBYRGJLXDUXB\",\"WARC-Block-Digest\":\"sha1:LJY3SAZIN4K3CF3NWUQVIJMFT2KC4WTM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510319.87_warc_CC-MAIN-20230927171156-20230927201156-00283.warc.gz\"}"}