1. Qpolynomial distanceregular graphs with a [sub] 1 [equal] 0 and a [sub] 2 [not equal] 0Štefko Miklavič, 2008, original scientific article Abstract: Let ▫$\Gamma$▫ denote a ▫$Q$▫polynomial distanceregular graph with diameter ▫$D \ge 3$▫ and intersection numbers ▫$a_1=0$▫, ▫$a_2 \ne 0$▫. Let ▫$X$▫ denote the vertex set of ▫$\Gamma$▫ and let ▫$A \in {\mathrm{Mat}}_X ({\mathbb{C}})$▫ denote the adjacency matrix of ▫$\Gamma$▫. Fix ▫$x \in X$▫ and let denote $A^\ast \in {\mathrm{Mat}}_X ({\mathbb{C}})$ the corresponding dual adjacency matrix. Let ▫$T$▫ denote the subalgebra of ▫$A{\mathrm{Mat}}_X ({\mathbb{C}})$▫ generated by ▫$A$▫, ▫$A^\ast$▫. We call ▫$T$▫ the Terwilliger algebra of ▫$\Gamma$▫ with respect to ▫$x$▫. We show that up to isomorphism there exists a unique irreducible ▫$T$▫module ▫$W$▫ with endpoint 1. We show that ▫$W$▫ has dimension ▫$2D2$▫. We display a basis for ▫$W$▫ which consists of eigenvectors for ▫$A^\ast$▫. We display the action of ▫$A$▫ on this basis. We show that ▫$W$▫ appears in the standard module of ▫$\Gamma$▫ with multiplicity ▫$k1$▫, where ▫$k$▫ is the valency of ▫$\Gamma$▫. Found in: ključnih besedah Summary of found: ...▫$A \in {\mathrm{Mat}}_X ({\mathbb{C}})$▫ denote the adjacency matrix of ▫$\Gamma$▫. Fix ▫$x \in X$▫ and... ...mathematics, graph theory, adjacency matrix, distanceregular graph, Terwilliger algebra... Keywords: mathematics, graph theory, adjacency matrix, distanceregular graph, Terwilliger algebra Published: 15.10.2013; Views: 1789; Downloads: 9 Full text (0,00 KB) 
2. Semisymmetric elementary abelian covers of the MöbiusKantor graphAleksander Malnič, Štefko Miklavič, Primož Potočnik, Dragan Marušič, 2007, original scientific article Abstract: Let ▫$\wp_N : \tilde{X} \to X$▫ be a regular covering projection of connected graphs with the group of covering transformations isomorphic to ▫$N$▫. If ▫$N$▫ is an elementary abelian ▫$p$▫group, then the projection ▫$\wp_N$▫ is called ▫$p$▫elementary abelian. The projection ▫$\wp_N$▫ is vertextransitive (edgetransitive) if some vertextransitive (edgetransitive) subgroup of Aut ▫$X$▫ lifts along ▫$\wp_N$▫, and semisymmetric if it is edge but not vertextransitive. The projection ▫$\wp_N$▫ is minimal semisymmetric if ▫$\wp_N$▫ cannot be written as a composition ▫$\wp_N = \wp \circ \wp_M$▫ of two (nontrivial) regular covering projections, where ▫$\pw_M$▫ is semisymmetric. Finding elementary abelian covering projections can be grasped combinatorially via a linear representation of automorphisms acting on the first homology group of the graph. The method essentially reduces to finding invariant subspaces of matrix groups over prime fields (see [A. Malnic, D. Marušic, P. Potocnik, Elementary abelian covers of graphs, J. Algebraic Combin. 20 (2004) 7197]). In this paper, all pairwise nonisomorphic minimal semisymmetric elementary abelian regular covering projections of the MöbiusKantor graph, the Generalized Petersen graph GP(8,3), are constructed. No such covers exist for ▫$p=2$▫. Otherwise, the number of such covering projections is equal to ▫$(p1)/4$▫ and ▫$1+(p1)/4$▫ in cases ▫$p \equiv 5,9,13,17,21 \pmod{24}$▫ and ▫$p \equiv 1 \pmod{24}$▫, respectively, and to ▫$(p+1)/4$▫ and ▫$1+(p+1)/4$▫ in cases ▫$p \equiv 3,7,11,15,23 \pmod{24}$▫ and ▫$p \equiv 19 \pmod{24}$▫, respectively. For each such covering projection the voltage rules generating the corresponding covers are displayed explicitly. Found in: ključnih besedah Summary of found: ...essentially reduces to finding invariant subspaces of matrix groups over prime fields (see [A. Malnic,... ...mathematics, graph theory, graph, covering projection, lifting automorphisms, homology group,... Keywords: mathematics, graph theory, graph, covering projection, lifting automorphisms, homology group, group representation, matrix group, invariant subspaces Published: 03.04.2017; Views: 931; Downloads: 57 Full text (0,00 KB) 
3. On maximal distances in a commuting graphBojan Kuzma, Polona Oblak, Gregor Dolinar, 2012, original scientific article Abstract: It is shown that matrices over algebraically closed fields that are farthest apart in the commuting graph must be nonderogatory. Rankone matrices and diagonalizable matrices are also characterized in terms of the commuting graph. Found in: ključnih besedah Summary of found: ...mathematics, linear algebra, graph theory, commuting graph, matrix algebra, algebraically closed field, centralizer, distance in... Keywords: matematika, linearna algebra, teorija grafov, komutirajoči grafi, matrična algebra, algebraično zaprt obseg, centralizator, razdalja v grafih, mathematics, linear algebra, graph theory, commuting graph, matrix algebra, algebraically closed field, centralizer, distance in graphs Published: 03.04.2017; Views: 904; Downloads: 134 Full text (0,00 KB) This document has more files! More...

4. Permanent versus determinant over a finite fieldGregor Dolinar, Aleksandr Èmilevič Guterman, Marko Orel, Bojan Kuzma, 2013, published scientific conference contribution Abstract: Let ▫$\mathbb{F}$▫ be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices ▫$\mathcal{H}_n(\mathbb{F})$▫ and for the whole matrix space ▫$M_n(\mathbb{F})$▫. It is known that for ▫$n = 2$▫, there are bijective linear maps ▫$\Phi$▫ on ▫$\mathcal{H}_n(\mathbb{F})$▫ and ▫$M_n(\mathbb{F})$▫ satisfying the condition per ▫$A = \det \Phi(A)$▫. As an application of the obtained results, we show that if ▫$n \ge 3$▫, then the situation is completely different and already for ▫$n = 3$▫, there is no pair ofmaps ▫$(\Phi, \phi)$▫, where ▫$\Phi$▫ is an arbitrary bijective map on matrices and ▫$\phi \colon \mathbb{F} \to \mathbb{F}$▫ is an arbitrary map such that per ▫$A = \phi(\det \Phi(A))$▫ for all matrices ▫$A$▫ from the spaces ▫$\mathcal{H}_n(\mathbb{F})$▫ and ▫$M_n(\mathbb{F})$▫, respectively. Moreover, for the space ▫$M_n(\mathbb{F})$▫, we show that such a pair of transformations does not exist also for an arbitrary ▫$n > 3$▫ if the field ▫$\mathbb{F}$▫ contains sufficiently many elements (depending on ▫$n$▫). Our results are illustrated by a number of examples. Found in: ključnih besedah Summary of found: ...Hermitian matrices ▫$\mathcal{H}_n(\mathbb{F})$▫ and for the whole matrix space ▫$M_n(\mathbb{F})$▫. It is known that for... ...mathematics, linear algebra, matrix theory, permanent, determinant... Keywords: mathematics, linear algebra, matrix theory, permanent, determinant Published: 03.04.2017; Views: 852; Downloads: 68 Full text (0,00 KB) 