Pub Date : 2021-09-27DOI: 10.5772/intechopen.98986
Elias Mwakilama, P. Ali, Patrick Chidzalo, Kambombo Mtonga, L. Eneya
Graph invariants such as distance have a wide application in life, in particular when networks represent scenarios in form of either a bipartite or non-bipartite graph. Average distance μ of a graph G is one of the well-studied graph invariants. The graph invariants are often used in studying efficiency and stability of networks. However, the concept of average distance in a neighborhood graph G′ and its application has been less studied. In this chapter, we have studied properties of neighborhood graph and its invariants and deduced propositions and proofs to compare radius and average distance measures between G and G′. Our results show that if G is a connected bipartite graph and G′ its neighborhood, then radG1′≤radG and radG2′≤radG whenever G1′ and G2′ are components of G′. In addition, we showed that radG′≤radG for all r≥1 whenever G is a connected non-bipartite graph and G′ its neighborhood. Further, we also proved that if G is a connected graph and G′ its neighborhood, then and μG1′≤μG and μG2′≤μG whenever G1′ and G2′ are components of G′. In order to make our claims substantial and determine graphs for which the bounds are best possible, we performed some experiments in MATLAB software. Simulation results agree very well with the propositions and proofs. Finally, we have described how our results may be applied in socio-epidemiology and ecology and then concluded with other proposed further research questions.
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Pub Date : 2021-08-07DOI: 10.5772/intechopen.98726
S. Monikandan
A graph is reconstructible if it is determined up to isomorphism from the collection of all its one-vertex deleted unlabeled subgraphs. One of the foremost unsolved problems in Graph Theory is the Reconstruction Conjecture, which asserts that every graph G on at least three vertices is reconstructible. In 1980’s, tremendous work was done and many significant results have been produced on the problem and its variations. During the last three decades, work on it has slowed down gradually. P. J. Kelly (1957) first noted that trees are reconstructible; but the proof is quite lengthy. A short proof, due to Greenwell and Hemminger (1973), was given which is based on a simple, but powerful, counting theorem. This chapter deals with the counting theorem and its subsequent applications; also it ends up with a reduction of the Reconstruction Conjecture using distance and connectedness, which may lead to the final solution of the conjecture.
如果图从其所有单顶点删除的未标记子图的集合确定为同构，则图是可重构的。图论中最重要的未解决问题之一是重构猜想，它断言至少三个顶点上的每个图G都是可重构的。20世纪80年代，人们对这一问题及其变化进行了大量的研究，取得了许多重要的成果。在过去的三十年里，它的工作逐渐放缓。P. J. Kelly(1957)首先指出树木是可重建的;但是证明相当冗长。Greenwell和Hemminger(1973)给出了一个简短的证明，它基于一个简单但强大的计数定理。本章讨论计数定理及其后续应用;最后利用距离和连通性对重构猜想进行约简，从而得到重构猜想的最终解。
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Pub Date : 2021-06-24DOI: 10.5772/intechopen.98592
Hanzhou Wu
Information hiding allows us to hide secret information into digital objects such as images without significantly distorting the objects. The object containing hidden information will be transmitted to a data receiver via a probably insecure channel. To securely transmit the object carrying hidden information, the distortion caused by data embedding should be as low as possible, which is referred to as the rate-distortion optimization problem. Many conventional methods optimize the data embedding procedure by a heuristic fashion, which may be not optimal in terms of the rate-distortion performance. In this chapter, we introduce novel approaches that use graph theory for information hiding. These graph models are general and can be used for improving the rate-distortion performance of information hiding systems. In addition to rate-distortion optimization, recent graph models used for system design of information hiding will be also reviewed. This chapter is intended as a tutorial introducing advanced graph models applied to information hiding.
{"title":"Graph Models in Information Hiding","authors":"Hanzhou Wu","doi":"10.5772/intechopen.98592","DOIUrl":"https://doi.org/10.5772/intechopen.98592","url":null,"abstract":"Information hiding allows us to hide secret information into digital objects such as images without significantly distorting the objects. The object containing hidden information will be transmitted to a data receiver via a probably insecure channel. To securely transmit the object carrying hidden information, the distortion caused by data embedding should be as low as possible, which is referred to as the rate-distortion optimization problem. Many conventional methods optimize the data embedding procedure by a heuristic fashion, which may be not optimal in terms of the rate-distortion performance. In this chapter, we introduce novel approaches that use graph theory for information hiding. These graph models are general and can be used for improving the rate-distortion performance of information hiding systems. In addition to rate-distortion optimization, recent graph models used for system design of information hiding will be also reviewed. This chapter is intended as a tutorial introducing advanced graph models applied to information hiding.","PeriodicalId":299884,"journal":{"name":"Graph Theory [Working Title]","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130828859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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