Pub Date : 2022-05-18DOI: 10.26493/1855-3974.2902.f01
Tahir Shamsher, S. Pirzada, M. Bhat
Let $Gamma=(G,sigma)$ be a signed graph, where $sigma$ is the sign function on the edges of $G$. In this paper, we use the operation of partial transpose to obtain non-isomorphic Laplacian cospectral signed graphs. We will introduce two new operations on signed graphs. These operations will establish a relationship between the adjacency spectrum of one signed graph with the Laplacian spectrum of another signed graph. As an application, these new operations will be utilized to construct several pairs of cospectral non-isomorphic signed graphs. Finally, we construct integral signed graphs.
{"title":"On adjacency and Laplacian cospectral non-isomorphic signed graphs","authors":"Tahir Shamsher, S. Pirzada, M. Bhat","doi":"10.26493/1855-3974.2902.f01","DOIUrl":"https://doi.org/10.26493/1855-3974.2902.f01","url":null,"abstract":"Let $Gamma=(G,sigma)$ be a signed graph, where $sigma$ is the sign function on the edges of $G$. In this paper, we use the operation of partial transpose to obtain non-isomorphic Laplacian cospectral signed graphs. We will introduce two new operations on signed graphs. These operations will establish a relationship between the adjacency spectrum of one signed graph with the Laplacian spectrum of another signed graph. As an application, these new operations will be utilized to construct several pairs of cospectral non-isomorphic signed graphs. Finally, we construct integral signed graphs.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82856327","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 : 2022-04-29DOI: 10.26493/1855-3974.2882.c5e
Stefan Hammer
We show that on cactus graphs the Szeged index is bounded above by twice the Wiener index. For the revised Szeged index the situation is reversed if the graph class is further restricted. Namely, if all blocks of a cactus graph are cycles, then its revised Szeged index is bounded below by twice its Wiener index. Additionally, we show that these bounds are sharp and examine the cases of equality. Along the way, we provide a formulation of the revised Szeged index as a sum over vertices, which proves very helpful, and may be interesting in other contexts.
{"title":"Comparing Wiener, Szeged and revised Szeged index on cactus graphs","authors":"Stefan Hammer","doi":"10.26493/1855-3974.2882.c5e","DOIUrl":"https://doi.org/10.26493/1855-3974.2882.c5e","url":null,"abstract":"We show that on cactus graphs the Szeged index is bounded above by twice the Wiener index. For the revised Szeged index the situation is reversed if the graph class is further restricted. Namely, if all blocks of a cactus graph are cycles, then its revised Szeged index is bounded below by twice its Wiener index. Additionally, we show that these bounds are sharp and examine the cases of equality. Along the way, we provide a formulation of the revised Szeged index as a sum over vertices, which proves very helpful, and may be interesting in other contexts.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"83 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73015826","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 : 2022-04-14DOI: 10.26493/1855-3974.2340.a61
N. Bushaw, Blake Conka, Vinay Gupta, Aidan Kierans, Hudson Lafayette, C. Larson, Kevin McCall, Andriy Mulyar, Christine Sullivan, Scott Taylor, Evan Wainright, E. Wilson, Guanyu Wu, Sarah Loeb
{"title":"Bootstrap percolation via automated conjecturing","authors":"N. Bushaw, Blake Conka, Vinay Gupta, Aidan Kierans, Hudson Lafayette, C. Larson, Kevin McCall, Andriy Mulyar, Christine Sullivan, Scott Taylor, Evan Wainright, E. Wilson, Guanyu Wu, Sarah Loeb","doi":"10.26493/1855-3974.2340.a61","DOIUrl":"https://doi.org/10.26493/1855-3974.2340.a61","url":null,"abstract":"","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"152 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77870497","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 : 2022-03-31DOI: 10.26493/1855-3974.2631.be0
Lingjuan Shi
Fullerene graph G is a connected plane cubic graph with only pentagonal and hexagonal faces, which is the molecular graph of carbon fullerene. A spanning subgraph of G is called a perfect star packing in G if its each component is isomorphic to K1,3. For an independent set D ⊆ V (G), if each vertex in V (G) D has exactly one neighbor in D, then D is called an efficient dominating set of G. In this paper we show that the number of vertices of a fullerene graph admitting a perfect star packing must be divisible by 8. This answers an open problem asked by Došlić et al. and also shows that a fullerene graph with an efficient dominating set has 8n vertices. In addition, we find some counterexamples for the necessity of Theorem 14 in [14] and list some subgraphs that preclude the existence of a perfect star packing of type P0.
{"title":"The fullerene graphs with a perfect star packing","authors":"Lingjuan Shi","doi":"10.26493/1855-3974.2631.be0","DOIUrl":"https://doi.org/10.26493/1855-3974.2631.be0","url":null,"abstract":"Fullerene graph G is a connected plane cubic graph with only pentagonal and hexagonal faces, which is the molecular graph of carbon fullerene. A spanning subgraph of G is called a perfect star packing in G if its each component is isomorphic to K1,3. For an independent set D ⊆ V (G), if each vertex in V (G) D has exactly one neighbor in D, then D is called an efficient dominating set of G. In this paper we show that the number of vertices of a fullerene graph admitting a perfect star packing must be divisible by 8. This answers an open problem asked by Došlić et al. and also shows that a fullerene graph with an efficient dominating set has 8n vertices. In addition, we find some counterexamples for the necessity of Theorem 14 in [14] and list some subgraphs that preclude the existence of a perfect star packing of type P0.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"101 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85812533","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 : 2022-03-30DOI: 10.26493/1855-3974.2857.07b
Michael Giudici, Eric Swartz
We study locally $s$-arc-transitive graphs arising from the quasiprimitive product action (PA). We prove that, for any locally $(G,2)$-arc-transitive graph with $G$ acting quasiprimitively with type PA on both $G$-orbits of vertices, the group $G$ does not act primitively on either orbit. Moreover, we construct the first examples of locally $s$-arc-transitive graphs of PA type that are not standard double covers of $s$-arc-transitive graphs of PA type, answering the existence question for these graphs.
{"title":"Locally s-arc-transitive graphs arising from product action","authors":"Michael Giudici, Eric Swartz","doi":"10.26493/1855-3974.2857.07b","DOIUrl":"https://doi.org/10.26493/1855-3974.2857.07b","url":null,"abstract":"We study locally $s$-arc-transitive graphs arising from the quasiprimitive product action (PA). We prove that, for any locally $(G,2)$-arc-transitive graph with $G$ acting quasiprimitively with type PA on both $G$-orbits of vertices, the group $G$ does not act primitively on either orbit. Moreover, we construct the first examples of locally $s$-arc-transitive graphs of PA type that are not standard double covers of $s$-arc-transitive graphs of PA type, answering the existence question for these graphs.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73350591","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 : 2022-03-28DOI: 10.26493/1855-3974.2704.31a
Yusuke Suzuki
{"title":"Finitizable set of reductions for polyhedral quadrangulations of closed surfaces","authors":"Yusuke Suzuki","doi":"10.26493/1855-3974.2704.31a","DOIUrl":"https://doi.org/10.26493/1855-3974.2704.31a","url":null,"abstract":"","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91282115","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 : 2022-03-25DOI: 10.26493/1855-3974.2930.8e4
Shao-Fei Du, Yao Tian, Hao Yu
After long term efforts, it was recently proved in cite{DKM2} that except for the Peterson graph, every connected vertex-transitive graph of order $rs$ has a Hamilton cycle, where $r$ and $s$ are primes. A natural topic is to solve the hamiltonian problem for connected vertex-transitive graphs of $2rs$. This topic is quite trivial, as the problem is still unsolved even for that of $r=3$. In this paper, it is shown that except for the Coxeter graph, every connected vertex-transitive graph of order $2rs$ contains a Hamilton cycle, provided the automorphism group acts primitively on vertices.
{"title":"Hamilton cycles in primitive graphs of order 2rs","authors":"Shao-Fei Du, Yao Tian, Hao Yu","doi":"10.26493/1855-3974.2930.8e4","DOIUrl":"https://doi.org/10.26493/1855-3974.2930.8e4","url":null,"abstract":"After long term efforts, it was recently proved in cite{DKM2} that except for the Peterson graph, every connected vertex-transitive graph of order $rs$ has a Hamilton cycle, where $r$ and $s$ are primes. A natural topic is to solve the hamiltonian problem for connected vertex-transitive graphs of $2rs$. This topic is quite trivial, as the problem is still unsolved even for that of $r=3$. In this paper, it is shown that except for the Coxeter graph, every connected vertex-transitive graph of order $2rs$ contains a Hamilton cycle, provided the automorphism group acts primitively on vertices.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77215968","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 : 2022-03-23DOI: 10.26493/1855-3974.2853.b51
Vesna Iršič, Marusa Lekse, Mihael Pacnik, Petra Podlogar, M. Pracek
After $2$-crossing-critical graphs were characterized in 2016, their most general subfamily, large $3$-connected $2$-crossing-critical graphs, has attracted separate attention. This paper presents sharp upper and lower bounds for their domination and independence number.
{"title":"Domination and independence number of large 2-crossing-critical graphs","authors":"Vesna Iršič, Marusa Lekse, Mihael Pacnik, Petra Podlogar, M. Pracek","doi":"10.26493/1855-3974.2853.b51","DOIUrl":"https://doi.org/10.26493/1855-3974.2853.b51","url":null,"abstract":"After $2$-crossing-critical graphs were characterized in 2016, their most general subfamily, large $3$-connected $2$-crossing-critical graphs, has attracted separate attention. This paper presents sharp upper and lower bounds for their domination and independence number.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73996089","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 : 2022-03-02DOI: 10.26493/1855-3974.2692.86d
A. Burgess, Nicholas J. Cavenagh, David A. Pike
An ${ell}$-cycle system ${mathcal F}$ of a graph $Gamma$ is a set of ${ell}$-cycles which partition the edge set of $Gamma$. Two such cycle systems ${mathcal F}$ and ${mathcal F}'$ are said to be {em orthogonal} if no two distinct cycles from ${mathcal F}cup {mathcal F}'$ share more than one edge. Orthogonal cycle systems naturally arise from face $2$-colourable polyehdra and in higher genus from Heffter arrays with certain orderings. A set of pairwise orthogonal $ell$-cycle systems of $Gamma$ is said to be a set of mutually orthogonal cycle systems of $Gamma$. Let $mu(ell,n)$ (respectively, $mu'(ell,n)$) be the maximum integer $mu$ such that there exists a set of $mu$ mutually orthogonal (cyclic) $ell$-cycle systems of the complete graph $K_n$. We show that if $ellgeq 4$ is even and $nequiv 1pmod{2ell}$, then $mu'(ell,n)$, and hence $mu(ell,n)$, is bounded below by a constant multiple of $n/ell^2$. In contrast, we obtain the following upper bounds: $mu(ell,n)leq n-2$; $mu(ell,n)leq (n-2)(n-3)/(2(ell-3))$ when $ell geq 4$; $mu(ell,n)leq 1$ when $ell>n/sqrt{2}$; and $mu'(ell,n)leq n-3$ when $n geq 4$. We also obtain computational results for small values of $n$ and $ell$.
{"title":"Mutually orthogonal cycle systems","authors":"A. Burgess, Nicholas J. Cavenagh, David A. Pike","doi":"10.26493/1855-3974.2692.86d","DOIUrl":"https://doi.org/10.26493/1855-3974.2692.86d","url":null,"abstract":"An ${ell}$-cycle system ${mathcal F}$ of a graph $Gamma$ is a set of ${ell}$-cycles which partition the edge set of $Gamma$. Two such cycle systems ${mathcal F}$ and ${mathcal F}'$ are said to be {em orthogonal} if no two distinct cycles from ${mathcal F}cup {mathcal F}'$ share more than one edge. Orthogonal cycle systems naturally arise from face $2$-colourable polyehdra and in higher genus from Heffter arrays with certain orderings. A set of pairwise orthogonal $ell$-cycle systems of $Gamma$ is said to be a set of mutually orthogonal cycle systems of $Gamma$. Let $mu(ell,n)$ (respectively, $mu'(ell,n)$) be the maximum integer $mu$ such that there exists a set of $mu$ mutually orthogonal (cyclic) $ell$-cycle systems of the complete graph $K_n$. We show that if $ellgeq 4$ is even and $nequiv 1pmod{2ell}$, then $mu'(ell,n)$, and hence $mu(ell,n)$, is bounded below by a constant multiple of $n/ell^2$. In contrast, we obtain the following upper bounds: $mu(ell,n)leq n-2$; $mu(ell,n)leq (n-2)(n-3)/(2(ell-3))$ when $ell geq 4$; $mu(ell,n)leq 1$ when $ell>n/sqrt{2}$; and $mu'(ell,n)leq n-3$ when $n geq 4$. We also obtain computational results for small values of $n$ and $ell$.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"219 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89378906","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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