[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-135387-en":3,"doc-seo-135387-105":31,"detail-sidebar-cat-0-en-105":93},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},135387,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","SOME CONTRIBUTIONS TO HUB PARAMETERS OF GRAPHS - A THESIS - Master Degree in Mathematics","A formal master’s thesis in mathematics investigates hub parameters defined for graphs, developing theory around the “hop hub number,” “hop hub-integrity,” and related concepts. The study introduces foundational definitions and terminology, presents background context, and outlines research objectives and methodology before deriving new results. Additional contributions establish results for splitting graphs and introduce the “hop hubtic number” and “hop hub polynomial,” including relationships for standard graphs.","SOME CONTRIBUTIONS TO HUB PARAMETERS OF GRAPHS  \nA THESIS  \nSUBMITTED AS PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF THE MASTER DEGREE IN  \nMATHEMATICS  \nAL-BAYDHA University  \n(Department of Mathematics)  \nBY  \nAbdu-Alkafi Saead Qaid Sanad  \nUNDER THE GUIDANCE OF  \nDr. Sultan Senan Mahde  \nFaculty of Education and Science-Rada'a Al-Baydha University  \nAl-Baydha, Yemen  \nMarch, 2022  \nDEDICATED  \nThis Thesis is lovingly dedicated to the dearest people of my  \nheart    \nTo the first teacher in life, source of all that is important tome, for Muhamad is messenger of god.  \nTo my first love, my mother.  \nfor years of unwavering support and guidance my beloved  \nfather.  \nTo the pure soul of my brother, the martyr muhamad.  \nTo my beloved, sisters and brothers.  \nTo my love and happiness, To my wife.  \nTo every piece of my soul, my children.  \nTo the best teacher in my life, source of my strengths,  \nTo the secret of my strength in life, all my friends.  \nTo the secret of my success all, my students.  \nTo all of them  \nI say,  \nI love you so much.  \nCERTIFICATE  \nThis is to certify that the thesis \"SOME CONTRIBUTIONS TO HUB PARAMETERS OF GRAPHS\" is the result of research work done by Mr Abdu-Alkafi Saead Qaid Sanad under my supervision.  \nThe thesis has been the outcome of the original investigation in his area of research for a period for a year. The thesis as a whole or any part of it has not been submitted in part or in full to any other institution for the award of any other degree/diploma/associate ship/fellowship.  \nSignature of the Research Supervisor and Date  \n(Dr. Sultan Senan Mahde)  \nAcknowledgments  \nPraise is to Allah, Lord of the Heavens, the Earth, who gave me the potential and ability to complete this thesis.  \nFirst of all, I would like to express my sincere gratitude to my Guide Dr. Sultan Senan Mahde, Assistant Professor, Department of Mathematics, Radaa - Al-Baydha University, for all the assistance he provided me, and for the guidance, and continuous encouragement greatly in reaching the end of this thesis. I have expended energy in shaping my research competence, politically reviewing my work and making valuable suggestions. And all the basic information from beginning to end is all thanks, appreciation and respect.  \nSecondly, I would like to thank every doctor who taught me or contributed to my teaching from university doctors in the introductory stage, especially Prof. Dr. Abdullah Al-Sherai. I also extend my full appreciation and thanks to the doctors and their support and encouragement to finish this work, and also thanks to the University Administration represented by the President of the University, Prof. Ahmed Al-Arami, for his effective contribution to the success and approval of the program.  \nI also thank my colleagues at work for providing all the support and continuous encouragement and solving all the difficulties I face.  \nMy special thanks and appreciation to my beloved wife, Mrs. A. Al-  \nHanomi, for her constant care and support. She has always made me feel that\"I could do anything that I want to achieve\", and my lovely daughter malakand my son omar, whose natural smile and compassion always extended tome the real relief.  \nFinally, my ineffable gratitude goes to my whole family, my parents, wife, children, brothers and sisters for all their love, support and patience. Without their support and encouragement, I cannot concentrate on my work and overcome the difficulties.  \nAbdu-Alkafi Saead Qaid Sanad  \nTable of Contents  \nTable of Contents v  \n1 Introduction xiii  \n1.1 Introduction ................................ 1  \n1.2 Basic definitions and terminologies of a graph ............. 4  \n1.3 Background of the study ......................... 19  \n1.4 Research Objectives ........................... 22  \n1.5 Methodology ............................... 23  \n1.6 Short outline of the thesis ........................ 24  \n2 Hop hub number of graphs 26  \n2.1 Introduction ...........................","cbCaidmlcM3C4gT9","https://ap.wps.com/l/cbCaidmlcM3C4gT9","pdf",1925247,5,1,107,"English","en",105,"# Introduction\n## Introduction\n## Basic definitions and terminologies of a graph\n## Background of the study\n## Research Objectives\n## Methodology\n## Short outline of the thesis\n# Hop hub number of graphs\n## Introduction\n## The hop hub number of graphs\n## Hop hub number of line graphs\n# Hop hub-integrity of graphs\n## Introduction\n## The hop hub-integrity of graphs\n## Some properties of hop hub-integrity of line graphs\n# New results on hop hub number and hop hub-integrity of splitting graph\n## Introduction\n## The hop hub number of splitting graph\n## The hop hub-integrity of splitting graph\n# Hop hubtic number and hop hub polynomial of graphs\n## Introduction\n## The hop hubtic number of graphs\n## Hop hub polynomial of graphs\n## Hop hub polynomial of standard graphs","[{\"question\":\"What mathematical objects does the thesis study?\",\"answer\":\"The thesis studies graphs and specific hub parameters defined on them, including hop hub number and hop hub-integrity.\"},{\"question\":\"What are the main research components covered?\",\"answer\":\"It covers introductions and basic graph definitions, then develops results for hop hub number, hop hub-integrity, and new findings for splitting graphs.\"},{\"question\":\"What additional hub-related quantities are introduced or analyzed?\",\"answer\":\"The thesis introduces the hop hubtic number and studies hop hub polynomials of graphs and standard graphs.\"}]","SOME CONTRIBUTIONS TO HUB PARAMETERS OF GRAPHS - A THESIS - Master Degree in Mathematics | PDF",1787310544,270,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":88,"head_meta":90,"extra_data":92,"updated_unix":29},"some-contributions-to-hub-parameters-of-graphs-a-thesis-master-degree-in-mathematics","",{"@graph":37,"@context":87},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/some-contributions-to-hub-parameters-of-graphs-a-thesis-master-degree-in-mathematics/135387/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-08-24","2026-08-21",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83],{"name":74,"@type":75,"acceptedAnswer":76},"What mathematical objects does the thesis study?","Question",{"text":77,"@type":78},"The thesis studies graphs and specific hub parameters defined on them, including hop hub number and hop hub-integrity.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"What are the main research components covered?",{"text":82,"@type":78},"It covers introductions and basic graph definitions, then develops results for hop hub number, hop hub-integrity, and new findings for splitting graphs.",{"name":84,"@type":75,"acceptedAnswer":85},"What additional hub-related quantities are introduced or analyzed?",{"text":86,"@type":78},"The thesis introduces the hop hubtic number and studies hop hub polynomials of graphs and standard graphs.","https://schema.org",{"og:url":53,"og:type":89,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":91,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":94},[95,99,103,107,111,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":100,"show_sort_weight":101,"slug":102},"Literature",80,"literature",{"id":54,"doc_module":4,"doc_module_name":47,"category_name":104,"show_sort_weight":105,"slug":106},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":20,"slug":139},19,"General","general"]