فهرست مطالب
Transactions on Combinatorics
Volume:8 Issue: 4, Dec 2019
- تاریخ انتشار: 1398/09/10
- تعداد عناوین: 5
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Pages 1-9A signed graph $(G,sigma)$ is a graph together with an assignment of signs ${+,-}$ to its edges where $sigma$ is the subset of its negative edges. There are a few variants of coloring and clique problems of signed graphs, which have been studied. An initial version known as vertex coloring of signed graphs is defined by Zaslavsky in $1982$. Recently Naserasr et. al., in [R. Naserasr, E. Rollova and E. Sopena, Homomorphisms of signed graphs, J. Graph Theory, 79 (2015) 178--212, have defined signed chromatic and signed clique numbers of signed graphs. In this paper we consider the latter mentioned problems for signed interval graphs. We prove that the coloring problem of signed interval graphs is NP-complete whereas their ordinary coloring problem (the coloring problem of interval graphs) is in P. Moreover we prove that the signed clique problem of a signed interval graph can be solved in polynomial time. We also consider the complexity of further related problems.Keywords: Signed clique Problem, Signed Interval Graphs, Signed Coloring Problem
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Pages 11-21
We consider some combinatorics of elliptic root systems of type $A_1$. In particular, with respect to a fixed reflectable base, we give a precise description of the positive roots in terms of a positivity theorem. Also the set of reduced words of the corresponding Weyl group is precisely described. These then lead to a new characterization of the core of the corresponding Lie algebra, namely we show that the core is generated by positive root spaces.
Keywords: Elliptic root systems, Elliptic Lie algebras, Jordan algebras -
Pages 23-33Let $mathcal{S}_q$ denote the group of all square elements in the multiplicative group $mathbb{F}_q^*$ of a finite field $mathbb{F}_q$ of odd characteristic containing $q$ elements. Let $mathcal{O}_q$ be the set of all odd order elements of $mathbb{F}_q^*$. Then $mathcal{O}_q$ turns up as a subgroup of $mathcal{S}_q$. In this paper, we show that $mathcal{O}_q=langle4rangle$ if $q=2t+1$ and, $mathcal{O}_q=langle trangle $ if $q=4t+1$, where $q$ and $t$ are odd primes. Further, we determine the coefficients of irreducible factors of $x^{2^nt}-1$ using generators of these special subgroups of $mathbb{F}_q^*$Keywords: Polynomials over finite fields, Cyclotomic polynomials, Special groups
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Pages 35-48The generalized Zagreb index is an extension of both ordinary and variable Zagreb indices. In this paper, we present exact formulae for the values of the generalized Zagreb index for product graphs. Results are applied to some graphs of general and chemical interest such as nanotubes and nanotori.Keywords: Vertex degree, graph operation, nanotube, nanotorus
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Pages 49-60
Let G=(V(G),E(G)) be a digraph without loops and multiarcs, where V(G)={v1,v2, …,vn} and E(G) are the vertex set and the arc set of G, respectively. Let d+i be the outdegree of the vertex vi. Let A(G) be the adjacency matrix of G and D(G)=diag(d+1,d+2,…,d+n) be the diagonal matrix with outdegrees of the vertices of G. Then we call Q(G)=D(G)+A(G) the signless Laplacian matrix of G. The spectral radius of Q(G) is called the signless Laplacian spectral radius of G, denoted by q(G). In this paper, some upper bounds for q(G) are obtained. Furthermore, some upper bounds on q(G) involving outdegrees and the average 2-outdegrees of the vertices of G are also derived.
Keywords: digraph, Signless Laplacian spectral radius, Upper bounds