15 research outputs found
Determinant Formulas of Some Hessenberg Matrices with Jacobsthal Entries
In this paper, we evaluate determinants of several families of Hessenberg matrices having various subsequences of the Jacobsthal sequence as their nonzero entries. These identities may be written equivalently as formulas for certain linearly recurrent sequences expressed in terms of sums of products of Jacobsthal numbers with multinomial coefficients. Among the sequences that arise in this way include the Mersenne, Lucas and Jacobsthal-Lucas numbers as well as the squares of the Jacobsthal and Mersenne sequences. These results are extended to Hessenberg determinants involving sequences that are derived from two general families of linear second-order recurrences. Finally, combinatorial proofs are provided for several of our determinant results which make use of various correspondences between Jacobsthal tilings and certain restricted classes of binary words
A Creative Review on Coprime (Prime) Graphs
Coprime labelings and Coprime graphs have been of interest since 1980s and got popularized by the Entringer-Tout Tree Conjecture. Around the same time Newman's coprime mapping conjecture was settled by Pomerance and Selfridge. This result was further extended to integers in arithmetic progression. Since then coprime graphs were studied for various combinatorial properties. Here, coprimality of graphs for classes of graphs under the themes: Bipartite with special attention to Acyclicity, Eulerian and Regularity. Extremal graphs under non-coprimality and Eulerian properties are studied. Embeddings of coprime graphs in the general graphs, the maximum coprime graph and the Eulerian coprime graphs are studied as subgraphs and induced subgraphs. The purpose of this review is to assimilate the available works on coprime graphs. The results in the context of these themes are reviewed including embeddings and extremal problems
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Mini-Workshop: Lattice Polytopes: Methods, Advances, Applications
Lattice polytopes arise naturally in many different branches of pure and applied mathematics such as number theory, commutative algebra, combinatorics, toric geometry, optimization, and mirror symmetry. The miniworkshop on “Lattice polytopes: methods, advances, applications” focused on two current hot topics: the classification of lattice polytopes with few lattice points and unimodality questions for Ehrhart polynomials. The workshop consisted of morning talks on recent breakthroughs and new methods, and afternoon discussion groups where participants from a variety of different backgrounds explored further applications, identified open questions and future research directions, discussed specific examples and conjectures, and collaboratively tackled open research problems
Product of digraphs, (super) edge-magic valences and related problems
Discrete Mathematics, and in particular Graph Theory, has gained a lot of popularity during the last 7 decades. Among the many branches in Graph Theory, graph labelings has experimented a fast development, in particular during the last decade. One of the very important type of labelings are super edge-magic labelings introduced in 1998 by Enomoto et al. as a particular case of edge-magic labelings, introduced in 1970 by Kotzig and Rosa. An edge-magic labeling is a bijective mapping from the set of vertices and edges to [1, |V(G)|+|E(G)|], such that the sum of the labels of each edge and the incident vertices to it is constant. The constant is called the valence of the labeling. The edge-magic labeling is called super edge-magic if the smallest labels are assigned to the vertices.
In this thesis, we consider three problems related to (super) edge-magic labelings and (di)graph products in which we use a family of super edge-magic digraphs as a second factor of the product. The digraph product we use, the h-product, was introduced by Figueroa-Centeno et al. in 2008. It is a generalization of the Kronecker product of digraphs.
In Chapter 2, we study the super edge-magicness of graphs of equal order and size either by providing super edge-magic labelings of some elements in the family or proving that these labelings do not exist. The negative results are specially interesting since these kind of results are not common in the literature. Furthermore, the few results found in this direction usually meet one of the following reasons: too many vertices compared with the number of edges; too many edges compared with the number of vertices; or parity conditions. In our case, all previous reasons fail.
In Chapter 3, we enlarge the family of perfect (super) edge-magic crowns. A crown is obtained from a cycle by adding the same number of pendant edges to each vertex of the cycle. Intuitively speaking, a (super) edge-magic graphs is perfect (super) edge-magic if all possible theoretical valences occur. The main result of the chapter is that the crowns defined by a cycle of length pq, where p and q are different odd primes, are perfect (super) edge-magic. We also provided lower bounds for the number of edge-magic valences of crowns.
For graphs of equal order and size, the odd and the even labelling construction allows to obtain two edge-magic labelings from a particular super edge-magic labeling. The name refers to the parity of the vertex labels. In Chapter 4, we begin by providing some properties of odd and even labelling construction related to the (super) edge-magic labeling and also with respect to the digraph product. We also get a new application of the h-product by interchanging the role of the factors. This allows us to consider the classical conjecture of Godbold and Slater with respect to valences of cycles with a different point of view than the ones existing.
Finally, we devote Chapter 5 to study the problem of edge-magic valences of crowns, in which even cycles appear, and to establish a relationship between super edge-magic graphs and graph decompositions. Some lower bounds on the number of (super) edge-magic valences are also established.La MatemĂ tica Discreta, i en particular la Teoria de Grafs, han guanyat molta popularitat durant les Ăşltimes set dècades. Entre les moltes branques de la Teoria de Grafs, els etiquetatges de grafs han experimentat un rĂ pid desenvolupament, especialment durant l'Ăşltima dècada. Un dels tipus d'etiquetatges mĂ©s importants sĂłn els etiquetatges super branca-mĂ gics introduĂŻts el 1998 per Enomoto et al. com un cas particular d'etiquetatges branca-mĂ gics, introduĂŻts el 1970 per Kotzig i Rosa. Un etiquetatge branca-mĂ gic Ă©s una aplicaciĂł bijectiva del conjunt de vèrtexs i branques a [1, |V(G)|+|E(G)|], de manera que la suma de les etiquetes de cada branca i els vèrtexs incidents a ella Ă©s constant. La constant s'anomena valència de l'etiquetatge. L'etiquetatge branca-mĂ gic s'anomena super branca-mĂ gic si les etiquetes mĂ©s petites s'assignen als vèrtexs. En aquesta tesi, considerem tres problemes relacionats amb etiquetatges (super) branca-mĂ gic i productes de digrafs, en els que intervĂ© una famĂlia de grafs super branca-mĂ gic com a segon factor del producte. El producte de digrafs que usem, el producte h, va ser introduĂŻt per Figueroa-Centeno et al. el 2008. És una generalitzaciĂł del producte de Kronecker de digraphs. En el CapĂtol 2, estudiem el carĂ cter super branca-mĂ gic de grafs d’ordre igual a mida, ja sigui proporcionant etiquetatges super branca-mĂ gics d'alguns elements de la famĂlia o demostrant que aquests tipus d’etiquetatges no existeixen. Els resultats negatius sĂłn especialment interessants ja que aquest tipus de resultats no sĂłn comuns en la literatura. A mĂ©s, els pocs resultats trobats en aquesta direcciĂł solen encabir-se en una de les raons segĂĽents: massa vèrtexs en comparaciĂł amb el nombre de branques; massa branques en comparaciĂł amb el nombre de vèrtexs; o condicions de paritat. En el nostre cas, totes les raons anteriors fracassen. En el CapĂtol 3, ampliem la famĂlia de corones (super) branca-mĂ giques perfectes. Una corona Ă©s el graf que s’obtĂ© a partir d’un afegint el mateix nombre de branques a cada vèrtex del cicle. IntuĂŻtivament parlant, un graf (super) branca mĂ gic Ă©s (super) branca mĂ gic si es donen totes les possibles valències teòriques. El resultat principal del capĂtol Ă©s que les corones definides per un cicle de longitud pq, on p i q sĂłn primers senars diferents, sĂłn (super) branca mĂ gics perfectes. TambĂ© proporcionem cotes inferiors per a la quantitat de valències mĂ giques de corones. Per a grafs d'igual ordre i mida, la construcciĂł de l'etiquetatge senar i parell permet obtenir dos etiquetatges branca-mĂ gics a partir d'un etiquetatge super branca-mĂ gic. El nom fa referència a la paritat de les etiquetes de vèrtex. Al capĂtol 4, comencem proporcionant algunes propietats de la construcciĂł de l'etiquetatge senar i parell relacionades amb l'etiquetatge (super) branca-mĂ gic del que proven i tambĂ© al producte h de dĂgrafs. TambĂ© obtenim una nova aplicaciĂł del producte h intercanviant el paper dels factors. Això ens permet considerar la conjectura de Godbold i Slater respecte a les valències dels cicles des d’un punt de vista diferent a les existents. Finalment, dediquem el CapĂtol 5 a estudiar el problema de les valències branca-mĂ giques de les corones, en les que apareixen cicles parells, i a establir una relaciĂł entre els grafs super branca-mĂ gic i les descomposicions de grafs. TambĂ© s'estableixen alguns cotes inferiors del nombre de valències (super) branca-mĂ giques.Postprint (published version
Product of digraphs, (super) edge-magic valences and related problems
Discrete Mathematics, and in particular Graph Theory, has gained a lot of popularity during the last 7 decades. Among the many branches in Graph Theory, graph labelings has experimented a fast development, in particular during the last decade. One of the very important type of labelings are super edge-magic labelings introduced in 1998 by Enomoto et al. as a particular case of edge-magic labelings, introduced in 1970 by Kotzig and Rosa. An edge-magic labeling is a bijective mapping from the set of vertices and edges to [1, |V(G)|+|E(G)|], such that the sum of the labels of each edge and the incident vertices to it is constant. The constant is called the valence of the labeling. The edge-magic labeling is called super edge-magic if the smallest labels are assigned to the vertices.
In this thesis, we consider three problems related to (super) edge-magic labelings and (di)graph products in which we use a family of super edge-magic digraphs as a second factor of the product. The digraph product we use, the h-product, was introduced by Figueroa-Centeno et al. in 2008. It is a generalization of the Kronecker product of digraphs.
In Chapter 2, we study the super edge-magicness of graphs of equal order and size either by providing super edge-magic labelings of some elements in the family or proving that these labelings do not exist. The negative results are specially interesting since these kind of results are not common in the literature. Furthermore, the few results found in this direction usually meet one of the following reasons: too many vertices compared with the number of edges; too many edges compared with the number of vertices; or parity conditions. In our case, all previous reasons fail.
In Chapter 3, we enlarge the family of perfect (super) edge-magic crowns. A crown is obtained from a cycle by adding the same number of pendant edges to each vertex of the cycle. Intuitively speaking, a (super) edge-magic graphs is perfect (super) edge-magic if all possible theoretical valences occur. The main result of the chapter is that the crowns defined by a cycle of length pq, where p and q are different odd primes, are perfect (super) edge-magic. We also provided lower bounds for the number of edge-magic valences of crowns.
For graphs of equal order and size, the odd and the even labelling construction allows to obtain two edge-magic labelings from a particular super edge-magic labeling. The name refers to the parity of the vertex labels. In Chapter 4, we begin by providing some properties of odd and even labelling construction related to the (super) edge-magic labeling and also with respect to the digraph product. We also get a new application of the h-product by interchanging the role of the factors. This allows us to consider the classical conjecture of Godbold and Slater with respect to valences of cycles with a different point of view than the ones existing.
Finally, we devote Chapter 5 to study the problem of edge-magic valences of crowns, in which even cycles appear, and to establish a relationship between super edge-magic graphs and graph decompositions. Some lower bounds on the number of (super) edge-magic valences are also established.La MatemĂ tica Discreta, i en particular la Teoria de Grafs, han guanyat molta popularitat durant les Ăşltimes set dècades. Entre les moltes branques de la Teoria de Grafs, els etiquetatges de grafs han experimentat un rĂ pid desenvolupament, especialment durant l'Ăşltima dècada. Un dels tipus d'etiquetatges mĂ©s importants sĂłn els etiquetatges super branca-mĂ gics introduĂŻts el 1998 per Enomoto et al. com un cas particular d'etiquetatges branca-mĂ gics, introduĂŻts el 1970 per Kotzig i Rosa. Un etiquetatge branca-mĂ gic Ă©s una aplicaciĂł bijectiva del conjunt de vèrtexs i branques a [1, |V(G)|+|E(G)|], de manera que la suma de les etiquetes de cada branca i els vèrtexs incidents a ella Ă©s constant. La constant s'anomena valència de l'etiquetatge. L'etiquetatge branca-mĂ gic s'anomena super branca-mĂ gic si les etiquetes mĂ©s petites s'assignen als vèrtexs. En aquesta tesi, considerem tres problemes relacionats amb etiquetatges (super) branca-mĂ gic i productes de digrafs, en els que intervĂ© una famĂlia de grafs super branca-mĂ gic com a segon factor del producte. El producte de digrafs que usem, el producte h, va ser introduĂŻt per Figueroa-Centeno et al. el 2008. És una generalitzaciĂł del producte de Kronecker de digraphs. En el CapĂtol 2, estudiem el carĂ cter super branca-mĂ gic de grafs d’ordre igual a mida, ja sigui proporcionant etiquetatges super branca-mĂ gics d'alguns elements de la famĂlia o demostrant que aquests tipus d’etiquetatges no existeixen. Els resultats negatius sĂłn especialment interessants ja que aquest tipus de resultats no sĂłn comuns en la literatura. A mĂ©s, els pocs resultats trobats en aquesta direcciĂł solen encabir-se en una de les raons segĂĽents: massa vèrtexs en comparaciĂł amb el nombre de branques; massa branques en comparaciĂł amb el nombre de vèrtexs; o condicions de paritat. En el nostre cas, totes les raons anteriors fracassen. En el CapĂtol 3, ampliem la famĂlia de corones (super) branca-mĂ giques perfectes. Una corona Ă©s el graf que s’obtĂ© a partir d’un afegint el mateix nombre de branques a cada vèrtex del cicle. IntuĂŻtivament parlant, un graf (super) branca mĂ gic Ă©s (super) branca mĂ gic si es donen totes les possibles valències teòriques. El resultat principal del capĂtol Ă©s que les corones definides per un cicle de longitud pq, on p i q sĂłn primers senars diferents, sĂłn (super) branca mĂ gics perfectes. TambĂ© proporcionem cotes inferiors per a la quantitat de valències mĂ giques de corones. Per a grafs d'igual ordre i mida, la construcciĂł de l'etiquetatge senar i parell permet obtenir dos etiquetatges branca-mĂ gics a partir d'un etiquetatge super branca-mĂ gic. El nom fa referència a la paritat de les etiquetes de vèrtex. Al capĂtol 4, comencem proporcionant algunes propietats de la construcciĂł de l'etiquetatge senar i parell relacionades amb l'etiquetatge (super) branca-mĂ gic del que proven i tambĂ© al producte h de dĂgrafs. TambĂ© obtenim una nova aplicaciĂł del producte h intercanviant el paper dels factors. Això ens permet considerar la conjectura de Godbold i Slater respecte a les valències dels cicles des d’un punt de vista diferent a les existents. Finalment, dediquem el CapĂtol 5 a estudiar el problema de les valències branca-mĂ giques de les corones, en les que apareixen cicles parells, i a establir una relaciĂł entre els grafs super branca-mĂ gic i les descomposicions de grafs. TambĂ© s'estableixen alguns cotes inferiors del nombre de valències (super) branca-mĂ giques
Errata and Addenda to Mathematical Constants
We humbly and briefly offer corrections and supplements to Mathematical
Constants (2003) and Mathematical Constants II (2019), both published by
Cambridge University Press. Comments are always welcome.Comment: 162 page