Let G be a finite group and let r ∈ N . An r -coloring of G is any mapping χ : G → { 1 , . . . , r } . A coloring χ is symmetric if there is g ∈ G such that χ ( gx − 1 g ) = χ ( x ) for every x ∈ G . We show that if f ( r ) is the polynomial representing the number of symmetric r -colorings of G , then the number of symmetric r -colorings of G × Z 2 is f ( r 2 )
{"title":"Symmetric colorings of G × Z_2","authors":"Jabulani Phakathi, Yevhen Zelenyuk, Yuliya Zelenyuk","doi":"10.5614/ejgta.2023.11.2.3","DOIUrl":"https://doi.org/10.5614/ejgta.2023.11.2.3","url":null,"abstract":"Let G be a finite group and let r ∈ N . An r -coloring of G is any mapping χ : G → { 1 , . . . , r } . A coloring χ is symmetric if there is g ∈ G such that χ ( gx − 1 g ) = χ ( x ) for every x ∈ G . We show that if f ( r ) is the polynomial representing the number of symmetric r -colorings of G , then the number of symmetric r -colorings of G × Z 2 is f ( r 2 )","PeriodicalId":43771,"journal":{"name":"Electronic Journal of Graph Theory and Applications","volume":"61 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135322963","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}
Pub Date : 2023-10-24DOI: 10.5614/ejgta.2023.11.2.14
Nobuaki Obata
We derive a formula for the QE constant of a complete multipartite graph and determine the complete multipartite graphs of non-QE class, namely, those which do not admit quadratic embeddings in a Euclidean space. Moreover, the primary non-QE graphs are specified among the complete multipartite graphs.
{"title":"Complete multipartite graphs of non-QE class","authors":"Nobuaki Obata","doi":"10.5614/ejgta.2023.11.2.14","DOIUrl":"https://doi.org/10.5614/ejgta.2023.11.2.14","url":null,"abstract":"We derive a formula for the QE constant of a complete multipartite graph and determine the complete multipartite graphs of non-QE class, namely, those which do not admit quadratic embeddings in a Euclidean space. Moreover, the primary non-QE graphs are specified among the complete multipartite graphs.","PeriodicalId":43771,"journal":{"name":"Electronic Journal of Graph Theory and Applications","volume":"15 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135322948","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}
Pub Date : 2023-10-24DOI: 10.5614/ejgta.2023.11.2.11
Djoko Suprijanto
We survey a recent progress of cyclic codes over finite rings and their generalization to skew cyclic as well as skew cyclic codes with derivation over finite rings, focusing on structural properties of the codes. We also report recent developments on the construction methods of linear codes from graphs, in particular strongly regular as well as distance regular graphs.
{"title":"Linear codes and cyclic codes over finite rings and their generalizations: a survey","authors":"Djoko Suprijanto","doi":"10.5614/ejgta.2023.11.2.11","DOIUrl":"https://doi.org/10.5614/ejgta.2023.11.2.11","url":null,"abstract":"We survey a recent progress of cyclic codes over finite rings and their generalization to skew cyclic as well as skew cyclic codes with derivation over finite rings, focusing on structural properties of the codes. We also report recent developments on the construction methods of linear codes from graphs, in particular strongly regular as well as distance regular graphs.","PeriodicalId":43771,"journal":{"name":"Electronic Journal of Graph Theory and Applications","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135321969","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}
Pub Date : 2023-10-24DOI: 10.5614/ejgta.2023.11.2.5
Mehrdad Nasernejad
Suppose that V = {1, …, n } is a non-empty set of n elements, S = { S 1 , …, S m } a non-empty set of m non-empty subsets of V . In this paper, by using some algebraic notions in commutative algebra, we investigate the question arises whether there exists an undirected finite simple graph G with V ( G )= V , where S is the set whose elements are the minimal dominating sets of G .
{"title":"An algebraic approach to sets defining minimal dominating sets of regular graphs","authors":"Mehrdad Nasernejad","doi":"10.5614/ejgta.2023.11.2.5","DOIUrl":"https://doi.org/10.5614/ejgta.2023.11.2.5","url":null,"abstract":"Suppose that V = {1, …, n } is a non-empty set of n elements, S = { S 1 , …, S m } a non-empty set of m non-empty subsets of V . In this paper, by using some algebraic notions in commutative algebra, we investigate the question arises whether there exists an undirected finite simple graph G with V ( G )= V , where S is the set whose elements are the minimal dominating sets of G .","PeriodicalId":43771,"journal":{"name":"Electronic Journal of Graph Theory and Applications","volume":"7 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135321967","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}
Pub Date : 2016-01-01DOI: 10.5614/ejgta.2016.4.2.5
S. U
This paper discusses the enumeration for rooted spanning trees and forests of the labelled join graphs Km +Hn and Km +Kn,p, where Hn is a graph with n isolated vertices.
{"title":"Enumeration for Spanning Trees and Forests of Join Graphs Based on the Combinatorial Decomposition","authors":"S. U","doi":"10.5614/ejgta.2016.4.2.5","DOIUrl":"https://doi.org/10.5614/ejgta.2016.4.2.5","url":null,"abstract":"This paper discusses the enumeration for rooted spanning trees and forests of the labelled join graphs Km +Hn and Km +Kn,p, where Hn is a graph with n isolated vertices.","PeriodicalId":43771,"journal":{"name":"Electronic Journal of Graph Theory and Applications","volume":"4 1","pages":"171 - 177"},"PeriodicalIF":0.7,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70731985","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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