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Luis Antonio De Almeida Vieira - One of the best experts on this subject based on the ideXlab platform.
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euclidean jordan algebras and some conditions over the spectra of a Strongly Regular Graph
4open, 2019Co-Authors: Luis Antonio De Almeida VieiraAbstract:Let G be a primitive Strongly Regular Graph G such that the Regularity is less than half of the order of G and A its matrix of adjacency, and let
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binomial hadamard series and inequalities over the spectra of a Strongly Regular Graph
Applied Mathematics-a Journal of Chinese Universities Series B, 2018Co-Authors: Luis Antonio De Almeida VieiraAbstract:Let G be a primitive Strongly Regular Graph of order n and A is adjacency matrix. In this paper we first associate to A a real 3-dimensional Euclidean Jordan algebra with rank three spanned by In and the natural powers of A that is a subalgebra of the Euclidean Jordan algebra of symmetric matrix of order n. Next we consider a basis that is a Jordan frame of . Finally, by an algebraic asymptotic analysis of the second spectral decomposition of some Hadamard series associated to A we establish some inequalities over the spectra and over the parameters of a Strongly Regular Graph.
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asymptotic properties of the spectra of a Strongly Regular Graph
International Conference on Innovation Engineering and Entrepreneurship, 2018Co-Authors: Luis Antonio De Almeida VieiraAbstract:Let G be a Strongly Regular Graph with three distinct eigenvalues and A his matrix of adjacency. In this work we associate a three dimensional real Euclidean Jordan algebra V with rank three to A and next we consider a Jordan frame B of idempotents of V. Next we analyse the spectra of a particular convergent Hadamard series of \(A^{2}\) and establish asymptotic inequalities over the spectra and the parameters of G.
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asymptotic inequalities on the parameters of a Strongly Regular Graph
AIP Conference Proceedings, 2017Co-Authors: Luis Antonio De Almeida VieiraAbstract:We first consider a Strongly Regular G whose adjacency matrix is A, next we associate a real three dimensional Euclidean Jordan algebra 𝒜 with rank three to the matrix A. Finally, from the analyze of the spectra of a binomial Hadamard Series of an element of 𝒜 we establish asymptotical inequalities on the parameters of a Strongly Regular Graph.
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Some results on the spectra of Strongly Regular Graphs
2016Co-Authors: Luis Antonio De Almeida Vieira, Vasco Moco ManoAbstract:Let G be a Strongly Regular Graph whose adjacency matrix is A. We associate a real finite dimensional Euclidean Jordan algebra 𝒱, of rank three to the Strongly Regular Graph G, spanned by I and the natural powers of A, endowed with the Jordan product of matrices and with the inner product as being the usual trace of matrices. Finally, by the analysis of the binomial Hadamard series of an element of 𝒱, we establish some inequalities on the parameters and on the spectrum of a Strongly Regular Graph like those established in theorems 3 and 4.
Vasco Moco Mano - One of the best experts on this subject based on the ideXlab platform.
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Some results on the spectra of Strongly Regular Graphs
2016Co-Authors: Luis Antonio De Almeida Vieira, Vasco Moco ManoAbstract:Let G be a Strongly Regular Graph whose adjacency matrix is A. We associate a real finite dimensional Euclidean Jordan algebra 𝒱, of rank three to the Strongly Regular Graph G, spanned by I and the natural powers of A, endowed with the Jordan product of matrices and with the inner product as being the usual trace of matrices. Finally, by the analysis of the binomial Hadamard series of an element of 𝒱, we establish some inequalities on the parameters and on the spectrum of a Strongly Regular Graph like those established in theorems 3 and 4.
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Generalized Krein Parameters of a Strongly Regular Graph
Applied Mathematics-a Journal of Chinese Universities Series B, 2015Co-Authors: Luis Antonio De Almeida Vieira, Vasco Moco ManoAbstract:We consider the real three-dimensional Euclidean Jordan algebra associated to a Strongly Regular Graph. Then, the Krein parameters of a Strongly Regular Graph are generalized and some generalized Krein admissibility conditions are deduced. Furthermore, we establish some relations between the classical Krein parameters and the generalized Krein parameters.
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Generalized Krein Parameters and Some Theorems on Strongly Regular Graphs
2015Co-Authors: Luis Antonio De Almeida Vieira, Vasco Moco ManoAbstract:Let G be a Strongly Regular Graph whose adjacency matrix is A. We associate a real finite dimensional Euclidean Jordan algebra 𝒜 of rank three to the Strongly Regular Graph G, endowed with the Jordan product of matrices and with the inner product as being the usual trace of matrices, spanned by I and the natural powers of A. Next we consider the unique Jordan frame ℒ associated to 𝒜 Finally, we define the generalized Krein parameters of G and establish some theorems on Strongly Regular Graphs.
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Inequalities on the Parameters of a Strongly Regular Graph
Springer Proceedings in Mathematics & Statistics, 2014Co-Authors: Vasco Moco Mano, Enide Andrade Martins, Luis Antonio De Almeida VieiraAbstract:In this paper we establish inequalities over the parameters and over the spectra of Strongly Regular Graphs in the environment of Euclidean Jordan algebras. We consider a Strongly Regular Graph, G, whose adjacency matrix A has three distinct eigenvalues, and the Euclidean Jordan algebra of real symmetric matrices of order n, \(\text{Sym}(n, \mathbb{R})\) with the vector product and the inner product being the Jordan product and the usual trace of matrices, respectively. We associate a three dimensional real Euclidean Jordan subalgebra \(\mathcal{A}\) of \(\text{Sym}(n, \mathbb{R})\) to A, spanned by the identity matrix and the natural powers of A. Next, we compute the unique complete system of orthogonal idempotents \(\mathcal{B}\) associated to A and we consider particular convergent Hadamard series constructed from the idempotents of \(\mathcal{B}\). Finally, by the analysis of the spectra of the sums of these Hadamard series we establish new conditions for the existence of a Strongly Regular Graph.
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euclidean jordan algebras maclaurin series and inequalities on Strongly Regular Graphs
11TH INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2013: ICNAAM 2013, 2013Co-Authors: Luis Antonio De Almeida Vieira, Vasco Moco ManoAbstract:Let X be a Strongly Regular Graph, whose adjacency matrix A has three distinct eigenvalues, and let V be the Euclidean Jordan algebra of real symmetric matrices, with the vector product and the inner product being the Jordan product and the usual trace of matrices, respectively. We consider the Euclidean Jordan subalgebra A of V spanned by the identity matrix and the natural powers of A. In this paper we work with the MacLaurin series of the sin function of some idempotents of A to obtain some inequalities over the parameters of X.
Andrea Švob - One of the best experts on this subject based on the ideXlab platform.
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on the mathrm psu 4 2 psu 4 2 invariant vertex transitive Strongly Regular 216 40 4 8 Graph
Graphs and Combinatorics, 2020Co-Authors: Dean Crnkovic, Francesco Pavese, Andrea ŠvobAbstract:In 2018 the first, Rukavina and the third author constructed with the aid of a computer the first example of a Strongly Regular Graph $$\Gamma$$ with parameters (216, 40, 4, 8) and proved that it is the unique $$\mathrm{PSU}(4,2)$$-invariant vertex-transitive Graph on 216 vertices. In this paper, using the geometry of the Hermitian surface of $$\mathrm{PG}(3,4)$$, we provide a computer-free proof of the existence of the Graph $$\Gamma$$. The maximal cliques of $$\Gamma$$ are also determined.
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on the psu 4 2 invariant vertex transitive Strongly Regular 216 40 4 8 Graph
arXiv: Combinatorics, 2019Co-Authors: Dean Crnkovic, Francesco Pavese, Andrea ŠvobAbstract:In 2018 the first, Rukavina and the third author constructed with the aid of a computer the first example of a Strongly Regular Graph $\Gamma$ with parameters (216, 40, 4, 8) and proved that it is the unique PSU(4,2)-invariant vertex-transitive Graph on 216 vertices. In this paper, using the geometry of the Hermitian surface of PG(3, 4), we provide a computer-free proof of the existence of the Graph $\Gamma$. The maximal cliques of $\Gamma$ are also determined.
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Strongly Regular Graphs from orthogonal groups $O^+(6,2)$ and $O^-(6,2)$
arXiv: Combinatorics, 2016Co-Authors: Dean Crnković, Sanja Rukavina, Andrea ŠvobAbstract:In this paper we construct all Strongly Regular Graphs, with at most 600 vertices, admitting a transitive action of the orthogonal group $O^+(6,2)$ or $O^-(6,2)$. Consequently, we prove the existence of Strongly Regular Graphs with parameters (216,40,4,8) and (540,187,58,68). We also construct a Strongly Regular Graph with parameters (540,224,88,96) that was to the best of our knowledge previously unknown. Further, we show that under certain conditions an orbit matrix $M$ of a Strongly Regular Graph $\Gamma$ can be used to define a new Strongly Regular Graph $\widetilde{\Gamma}$, where the vertices of the Graph $\widetilde{\Gamma}$ correspond to the orbits of $\Gamma$ (the rows of $M$). We show that some of the obtained Graphs are related to each other in a way that one can be constructed from an orbit matrix of the other.
A. A. Makhnev - One of the best experts on this subject based on the ideXlab platform.
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On automorphisms of a Strongly Regular Graph with parameters (210,95,40,45)
Proceedings of the Steklov Institute of Mathematics, 2012Co-Authors: A. A. Makhnev, N. V. ChuksinaAbstract:Possible orders of automorphisms of a Strongly Regular Graph with parameters (210,95,40,45) and the structure of fixed-point subGraphs of these automorphisms are found. © 2012 Pleiades Publishing, Ltd
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On automorphisms of a Strongly Regular Graph with parameters (210,95,40,45)
2012Co-Authors: A. A. Makhnev, N. V. ChuksinaAbstract:Possible orders of automorphisms of a Strongly Regular Graph with parameters (210,95,40,45) and the structure of fixed-point subGraphs of these automorphisms are found.
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On automorphisms of Strongly Regular Graphs with $ \lambda=0$, $ \mu=2$
Sbornik: Mathematics, 2004Co-Authors: A. A. Makhnev, V V NosovAbstract:The structure of fixed-point subGraphs of automorphisms of order 3 of Strongly Regular Graphs with parameters is determined. Let be the automorphism group of a hypothetical Strongly Regular Graph with parameters . Possible orders are found and the structure of fixed-point subGraphs is determined for elements of prime order in . The four-subgroups of are classified and the possible structure of the group is determined. A strengthening of a result of Nakagawa on the automorphism groups of Strongly Regular Graphs with , is obtained.
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On the Nonexistence of Strongly Regular Graphs with Parameters (486, 165, 36, 66)
Ukrainian Mathematical Journal, 2002Co-Authors: A. A. MakhnevAbstract:We prove that a Strongly Regular Graph with parameters (486, 165, 36, 66) does not exist. Since the parameters indicated are parameters of a pseudogeometric Graph for pG2(5, 32), we conclude that the partial geometries pG2(5, 32) and pG2(32, 5) do not exist. Finally, a neighborhood of an arbitrary vertex of a pseudogeometric Graph for pG3(6, 80) is a pseudogeometric Graph for pG2(5, 32) and, therefore, a pseudogeometric Graph for the partial geometry pG3(6, 80) [i.e., a Strongly Regular Graph with parameters (1127, 486, 165, 243)] does not exist.
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On Strongly Regular Graphs with k=2μ and Their Extensions
Siberian Mathematical Journal, 2002Co-Authors: A. A. MakhnevAbstract:We obtain a convenient expression for the parameters of a Strongly Regular Graph with k=2μ in terms of the nonprincipal eigenvalues x and −y. It turns out in particular that such Graphs are pseudogeometric for pGx(2x,y−1). We prove that a Strongly Regular Graph with parameters (35,16,6,8) is a quotient of the Johnson Graph \(\bar J\)(8,4). We also find the parameters of Strongly Regular Graphs in which the neighborhoods of vertices are pseudogeometric Graphs for pGx(2x,t),x≤3. In consequence, we establish that a connected Graph in which the neighborhoods of vertices are pseudogeometric Graphs for pG3(6,2) is isomorphic to the Taylor Graph on 72 vertices or to the alternating form Graph Alt(4,2) with parameters (64,35,18,20).
N. V. Chuksina - One of the best experts on this subject based on the ideXlab platform.
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On automorphisms of a Strongly Regular Graph with parameters (210,95,40,45)
2012Co-Authors: A. A. Makhnev, N. V. ChuksinaAbstract:Possible orders of automorphisms of a Strongly Regular Graph with parameters (210,95,40,45) and the structure of fixed-point subGraphs of these automorphisms are found.
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On automorphisms of a Strongly Regular Graph with parameters (210,95,40,45)
Proceedings of the Steklov Institute of Mathematics, 2012Co-Authors: A. A. Makhnev, N. V. ChuksinaAbstract:Possible orders of automorphisms of a Strongly Regular Graph with parameters (210,95,40,45) and the structure of fixed-point subGraphs of these automorphisms are found. © 2012 Pleiades Publishing, Ltd
