Pub Date : 2021-09-01DOI: 10.22108/TOC.2021.126646.1798
H. Topcu
$Kite_{n,p}$ denotes the kite graph that is obtained by appending complete graph with order $pgeq4$ to an endpoint of path graph with order $n-p$. It is shown that $Kite_{n,p}$ is determined by its adjacency spectrum for all $p$ and $n$ [H. Topcu and S. Sorgun, The kite graph is determined by its adjacency spectrum, Applied Math. and Comp., 330 (2018) 134--142]. For $n-p=1$, it is proven that $Kite_{n,p}$ is determined by its signless Laplacian spectrum when $ngeq4$, $nneq5$ and is also determined by its distance spectrum when $ngeq4$ [K. C. Das and M. Liu, Kite graphs are determined by their spectra, Applied Math. and Comp., 297 (2017) 74--78]. In this note, we say that $Kite_{n,p}$ is determined by its Laplacian spectrum for $n-pleq2$.
{"title":"$Kite_{p+2,p}$ is determined by its Laplacian spectrum","authors":"H. Topcu","doi":"10.22108/TOC.2021.126646.1798","DOIUrl":"https://doi.org/10.22108/TOC.2021.126646.1798","url":null,"abstract":"$Kite_{n,p}$ denotes the kite graph that is obtained by appending complete graph with order $pgeq4$ to an endpoint of path graph with order $n-p$. It is shown that $Kite_{n,p}$ is determined by its adjacency spectrum for all $p$ and $n$ [H. Topcu and S. Sorgun, The kite graph is determined by its adjacency spectrum, Applied Math. and Comp., 330 (2018) 134--142]. For $n-p=1$, it is proven that $Kite_{n,p}$ is determined by its signless Laplacian spectrum when $ngeq4$, $nneq5$ and is also determined by its distance spectrum when $ngeq4$ [K. C. Das and M. Liu, Kite graphs are determined by their spectra, Applied Math. and Comp., 297 (2017) 74--78]. In this note, we say that $Kite_{n,p}$ is determined by its Laplacian spectrum for $n-pleq2$.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"10 1","pages":"165-170"},"PeriodicalIF":0.4,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48957890","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 : 2021-08-31DOI: 10.22108/TOC.2021.127167.1816
H. Ramane, Daneshwari Patil, Kartik S. Pise
The energy of a graph is the sum of the absolute values of the eigenvalues of a graph. Two graphs are said to be equienergetic if they have same energy. A graph is said to be complementary equienergetic if it is equienergetic with its complement. Recently several complementary equienergetic graphs have been identified. In this paper, we characterize the cycle, path, complete bipartite regular graph and iterated line graph of regular graph, which are complementary equienergetic.
{"title":"Certain classes of complementary equienergetic graphs","authors":"H. Ramane, Daneshwari Patil, Kartik S. Pise","doi":"10.22108/TOC.2021.127167.1816","DOIUrl":"https://doi.org/10.22108/TOC.2021.127167.1816","url":null,"abstract":"The energy of a graph is the sum of the absolute values of the eigenvalues of a graph. Two graphs are said to be equienergetic if they have same energy. A graph is said to be complementary equienergetic if it is equienergetic with its complement. Recently several complementary equienergetic graphs have been identified. In this paper, we characterize the cycle, path, complete bipartite regular graph and iterated line graph of regular graph, which are complementary equienergetic.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41743677","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 : 2021-06-08DOI: 10.22108/TOC.2021.127880.1832
A. Mofidi
In this paper, we delve into studying some relations between the structure of the cycles and spanning trees of a graph through the lens of its cycle and spanning tree hypergraphs which are hypergraphs with the edge set of the graph as their vertices and the edge sets of the cycles and spanning trees as their hyperedges respectively. In particular, we investigate relations between these hypergraphs from the perspective of the VC-dimension and some important separating and covering features of hypergraph theory and amongst the results, for example show that the VC-dimension of the cycle hypergraph is less than or equal to the VC-dimension of the spanning tree hypergraph and their gap can be arbitrary large. Note that VC-dimension is an important measure of complexity and a fundamental notion in numerous fields such as extremal combinatorics, graph theory, statistics and the theory of machine learning. Also we compare the separating and covering features of the mentioned hypergraphs and for instance show that the separating number of the cycle hypergraph is less than or equal to that of the spanning tree hypergraph. These hypergraphs help us to make several connections between cycles and spanning trees of graphs and compare their complexities.
{"title":"On the VC-dimension, covering and separating properties of the cycle and spanning tree hypergraphs of graphs","authors":"A. Mofidi","doi":"10.22108/TOC.2021.127880.1832","DOIUrl":"https://doi.org/10.22108/TOC.2021.127880.1832","url":null,"abstract":"In this paper, we delve into studying some relations between the structure of the cycles and spanning trees of a graph through the lens of its cycle and spanning tree hypergraphs which are hypergraphs with the edge set of the graph as their vertices and the edge sets of the cycles and spanning trees as their hyperedges respectively. In particular, we investigate relations between these hypergraphs from the perspective of the VC-dimension and some important separating and covering features of hypergraph theory and amongst the results, for example show that the VC-dimension of the cycle hypergraph is less than or equal to the VC-dimension of the spanning tree hypergraph and their gap can be arbitrary large. Note that VC-dimension is an important measure of complexity and a fundamental notion in numerous fields such as extremal combinatorics, graph theory, statistics and the theory of machine learning. Also we compare the separating and covering features of the mentioned hypergraphs and for instance show that the separating number of the cycle hypergraph is less than or equal to that of the spanning tree hypergraph. These hypergraphs help us to make several connections between cycles and spanning trees of graphs and compare their complexities.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44787995","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 : 2021-06-01DOI: 10.22108/TOC.2021.123661.1739
Shahrooz Janbaz, Bagher Bagherpour, A. Zaghian
The characterization of the ideal access structures is one of the main open problems in secret sharing and is important from both practical and theoretical points of views. A graph-based $3-$homogeneous access structure is an access structure in which the participants are the vertices of a connected graph and every subset of the vertices is a minimal qualified subset if it has three vertices and induces a connected graph. In this paper, we introduce the graph-based $3-$homogeneous access structures and characterize the ideal graph-based $3$-homogeneous access structures. We prove that for every non-ideal graph-based $3$-homogeneous access structure over the graph $G$ with the maximum degree $d$ there exists a secret sharing scheme with an information rate $frac{1}{d+1}$. Furthermore, we mention three forbidden configurations that are useful in characterizing other families of ideal access structures.
{"title":"Ideal secret sharing schemes on graph-based $3$-homogeneous access structures","authors":"Shahrooz Janbaz, Bagher Bagherpour, A. Zaghian","doi":"10.22108/TOC.2021.123661.1739","DOIUrl":"https://doi.org/10.22108/TOC.2021.123661.1739","url":null,"abstract":"The characterization of the ideal access structures is one of the main open problems in secret sharing and is important from both practical and theoretical points of views. A graph-based $3-$homogeneous access structure is an access structure in which the participants are the vertices of a connected graph and every subset of the vertices is a minimal qualified subset if it has three vertices and induces a connected graph. In this paper, we introduce the graph-based $3-$homogeneous access structures and characterize the ideal graph-based $3$-homogeneous access structures. We prove that for every non-ideal graph-based $3$-homogeneous access structure over the graph $G$ with the maximum degree $d$ there exists a secret sharing scheme with an information rate $frac{1}{d+1}$. Furthermore, we mention three forbidden configurations that are useful in characterizing other families of ideal access structures.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"10 1","pages":"107-120"},"PeriodicalIF":0.4,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49391948","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 : 2021-06-01DOI: 10.22108/TOC.2020.125047.1764
S. Balachandran, T. Vetrík
Cruz, Monsalve and Rada [Extremal values of vertex-degree-based topological indices of chemical trees, Appl. Math. Comput. 380 (2020) 125281] posed an open problem to find the maximum value of the exponential second Zagreb index for chemical trees of given order. In this paper, we solve the open problem completely.
{"title":"Exponential second Zagreb index of chemical trees","authors":"S. Balachandran, T. Vetrík","doi":"10.22108/TOC.2020.125047.1764","DOIUrl":"https://doi.org/10.22108/TOC.2020.125047.1764","url":null,"abstract":"Cruz, Monsalve and Rada [Extremal values of vertex-degree-based topological indices of chemical trees, Appl. Math. Comput. 380 (2020) 125281] posed an open problem to find the maximum value of the exponential second Zagreb index for chemical trees of given order. In this paper, we solve the open problem completely.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"10 1","pages":"97-106"},"PeriodicalIF":0.4,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44708757","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 : 2021-06-01DOI: 10.22108/TOC.2021.124678.1758
V. Nourozi, Saeed Tafazolian, Farhad Rahamti
In this paper, we compute a formula for the $a$-number of certain maximal curves given by the equation $y^{q}+y=x^{frac{q+1}{2}}$ over the finite field $mathbb{F}_{q^2}$. The same problem is studied for the maximal curve corresponding to $sum_{t=1}^s y^{q/2^t}=x^{q+1}$ with $q=2^s$, over the finite field $mathbb{F}_{q^2}$.
{"title":"The $a$-number of jacobians of certain maximal curves","authors":"V. Nourozi, Saeed Tafazolian, Farhad Rahamti","doi":"10.22108/TOC.2021.124678.1758","DOIUrl":"https://doi.org/10.22108/TOC.2021.124678.1758","url":null,"abstract":"In this paper, we compute a formula for the $a$-number of certain maximal curves given by the equation $y^{q}+y=x^{frac{q+1}{2}}$ over the finite field $mathbb{F}_{q^2}$. The same problem is studied for the maximal curve corresponding to $sum_{t=1}^s y^{q/2^t}=x^{q+1}$ with $q=2^s$, over the finite field $mathbb{F}_{q^2}$.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"10 1","pages":"121-128"},"PeriodicalIF":0.4,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46024780","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 : 2021-03-01DOI: 10.22108/TOC.2020.123414.1733
Omer Egecloglu
We study two foremost Mahonian statistics, the major index and the inversion number for a class of binary words called restricted Fibonacci words. The language of restricted Fibonacci words satisfies recurrences which allow for the calculation of the generating functions in two different ways. These yield identities involving the $q$-binomial coefficients and provide non-standard $q$-analogues of the Fibonacci numbers. The major index generating function for restricted Fibonacci words turns out to be a $q$-power multiple of the inversion generating function.
{"title":"Statistics on restricted Fibonacci words","authors":"Omer Egecloglu","doi":"10.22108/TOC.2020.123414.1733","DOIUrl":"https://doi.org/10.22108/TOC.2020.123414.1733","url":null,"abstract":"We study two foremost Mahonian statistics, the major index and the inversion number for a class of binary words called restricted Fibonacci words. The language of restricted Fibonacci words satisfies recurrences which allow for the calculation of the generating functions in two different ways. These yield identities involving the $q$-binomial coefficients and provide non-standard $q$-analogues of the Fibonacci numbers. The major index generating function for restricted Fibonacci words turns out to be a $q$-power multiple of the inversion generating function.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"10 1","pages":"31-42"},"PeriodicalIF":0.4,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45843922","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 : 2021-03-01DOI: 10.22108/TOC.2020.121940.1715
A. Alhevaz, M. Baghipur, S. Pirzada, Y. Shang
Given a simple graph $G$, the distance signlesss Laplacian $D^{Q}(G)=Tr(G)+D(G)$ is the sum of vertex transmissions matrix $Tr(G)$ and distance matrix $D(G)$. In this paper, thanks to the symmetry of $D^{Q}(G)$, we obtain novel sharp bounds on the distance signless Laplacian eigenvalues of $G$, and in particular the distance signless Laplacian spectral radius. The bounds are expressed through graph diameter, vertex covering number, edge covering number, clique number, independence number, domination number as well as extremal transmission degrees. The graphs achieving the corresponding bounds are delineated. In addition, we investigate the distance signless Laplacian spectrum induced by Indu-Bala product, Cartesian product as well as extended double cover graph.
{"title":"Some inequalities involving the distance signless Laplacian eigenvalues of graphs","authors":"A. Alhevaz, M. Baghipur, S. Pirzada, Y. Shang","doi":"10.22108/TOC.2020.121940.1715","DOIUrl":"https://doi.org/10.22108/TOC.2020.121940.1715","url":null,"abstract":"Given a simple graph $G$, the distance signlesss Laplacian $D^{Q}(G)=Tr(G)+D(G)$ is the sum of vertex transmissions matrix $Tr(G)$ and distance matrix $D(G)$. In this paper, thanks to the symmetry of $D^{Q}(G)$, we obtain novel sharp bounds on the distance signless Laplacian eigenvalues of $G$, and in particular the distance signless Laplacian spectral radius. The bounds are expressed through graph diameter, vertex covering number, edge covering number, clique number, independence number, domination number as well as extremal transmission degrees. The graphs achieving the corresponding bounds are delineated. In addition, we investigate the distance signless Laplacian spectrum induced by Indu-Bala product, Cartesian product as well as extended double cover graph.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"10 1","pages":"9-29"},"PeriodicalIF":0.4,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42622076","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 : 2021-03-01DOI: 10.22108/TOC.2020.123692.1740
Xavier Mbaale, B. Rodrigues
Let $G = PSL_{2}(q)$, where $q$ is a power of an odd prime. Let $M$ be a maximal subgroup of $G$. Define $leftlbrace frac{|M|}{|M cap M^g|}: g in G rightrbrace$ to be the set of orbit lengths of the primitive action of $G$ on the conjugates of a maximal subgroup $M$ of $G.$ By using a method described by Key and Moori in the literature, we construct all primitive symmetric $1$-designs that admit $G$ as a permutation group of automorphisms.
设$G = PSL_{2}(q)$ $，其中$q$是奇素数的幂。设$M$是$G$的极大子群。定义$左括号frac{|M|}{|M cap M^g|}}}:} g中的$右括号$是$ g $在$ g的极大子群$M$共轭上的基元作用的轨道长度的集合。利用Key和Moori在文献中描述的方法，我们构造了所有承认$G$为自同构置换群的原始对称$1$-设计。
{"title":"Symmetric $1$-designs from $PSL_{2}(q),$ for $q$ a power of an odd prime","authors":"Xavier Mbaale, B. Rodrigues","doi":"10.22108/TOC.2020.123692.1740","DOIUrl":"https://doi.org/10.22108/TOC.2020.123692.1740","url":null,"abstract":"Let $G = PSL_{2}(q)$, where $q$ is a power of an odd prime. Let $M$ be a maximal subgroup of $G$. Define $leftlbrace frac{|M|}{|M cap M^g|}: g in G rightrbrace$ to be the set of orbit lengths of the primitive action of $G$ on the conjugates of a maximal subgroup $M$ of $G.$ By using a method described by Key and Moori in the literature, we construct all primitive symmetric $1$-designs that admit $G$ as a permutation group of automorphisms.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"10 1","pages":"43-61"},"PeriodicalIF":0.4,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46089321","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 : 2021-03-01DOI: 10.22108/TOC.2020.122877.1726
E. Milovanovic, M. Matejic, I. Milovanovic
Let $G=(V,E)$ be a simple graph with $nge 3$ vertices, $m$ edges and vertex degree sequence $Delta=d_1 ge d_2 ge cdots ge d_n=delta>0$. Denote by $S={1, 2,ldots,n}$ an index set and by $J={I=(r_1, r_2,ldots,r_k) , | , 1le r_1
设$G=（V，E）$是一个具有$nge 3$顶点的简单图, $m$边缘 顶点度序列$Delta＝d_1 ge d_2 ge cdots ge d_n=delta>0$. 表示为$S＝｛1，2，ldots，n｝$一个索引集，表示为 $J={I=（r_1，r_2，ldots，r_k）, | , 1le r_1
{"title":"Some remarks on the sum of powers of the degrees of graphs","authors":"E. Milovanovic, M. Matejic, I. Milovanovic","doi":"10.22108/TOC.2020.122877.1726","DOIUrl":"https://doi.org/10.22108/TOC.2020.122877.1726","url":null,"abstract":"Let $G=(V,E)$ be a simple graph with $nge 3$ vertices, $m$ edges and vertex degree sequence $Delta=d_1 ge d_2 ge cdots ge d_n=delta>0$. Denote by $S={1, 2,ldots,n}$ an index set and by $J={I=(r_1, r_2,ldots,r_k) , | , 1le r_1<r_2<cdots<r_kle n}$ a set of all subsets of $S$ of cardinality $k$, $1le kle n-1$. In addition, denote by $d_{I}=d_{r_1}+d_{r_2}+cdots+d_{r_k}$, $1le kle n-1$, $1le r_1<r_2<cdots<r_kle n-1$, the sum of $k$ arbitrary vertex degrees, where $Delta_{I}=d_{1}+d_{2}+cdots+d_{k}$ and $delta_{I}=d_{n-k+1}+d_{n-k+2}+cdots+d_{n}$. We consider the following graph invariant $S_{alpha,k}(G)=sum_{Iin J}d_I^{alpha}$, where $alpha$ is an arbitrary real number, and establish its bounds. A number of known bounds for various topological indices are obtained as special cases.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"10 1","pages":"63-71"},"PeriodicalIF":0.4,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48821606","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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