The relation between equiangular sets of lines in the real space and distance-regular double covers of the complete graph is well known and studied since the work of Seidel and others in the 70s. The ...main topic of this paper is to continue the study on how complex equiangular lines relate to distance-regular covers of the complete graph with larger index. Given a set of equiangular lines meeting the relative (or Welch) bound, we show that if the entries of the corresponding Gram matrix are prime roots of unity, then these lines can be used to construct an antipodal distance-regular graph of diameter three. We also study in detail how the absolute (or Gerzon) bound for a set of equiangular lines can be used to derive bounds of the parameters of abelian distance-regular covers of the complete graph.
Mathematical approaches to structure model problems have a significant role in expanding our knowledge in our routine life circumstances. To put them into practice, the right formulation, method, ...systematic representation, and formulation are needed. The purpose of introducing soft graphs is to discretize these fundamental mathematical ideas, which are inherently continuous, and to provide new tools for applying mathematical analysis technology to real-world applications including imperfect and inexact data or uncertainty. Soft rough covering models $ \left(\text{briefly}, \text{ }\mathcal{SRC}\text{-Models}\right) $, a novel theory that addresses uncertainty. In this present paper, we have introduced two new concepts $ \mathcal{L}\mathfrak{i} $-soft rough covering graphs ($ \mathcal{L}\mathfrak{i} $-$ \mathcal{SRCG} $s) and the concept of fixed point of such graphs. Furthermore, we looked into a some algebras that dealt with the fixed points of $ \mathcal{L}\mathfrak{i} $-$ \mathcal{SRCG} $s. Applications of the algebraic structures available in covering soft sets to soft graphs may reveal new facets of graph theory.
In this article, we introduce a geometric and a spectral preorder relation on the class of weighted graphs with a magnetic potential. The first preorder is expressed through the existence of a graph ...homomorphism respecting the magnetic potential and fulfilling certain inequalities for the weights. The second preorder refers to the spectrum of the associated Laplacian of the magnetic weighted graph. These relations give a quantitative control of the effect of elementary and composite perturbations of the graph (deleting edges, contracting vertices, etc.) on the spectrum of the corresponding Laplacians, generalising interlacing of eigenvalues. We give several applications of the preorders: we show how to classify graphs according to these preorders and we prove the stability of certain eigenvalues in graphs with a maximal
d
-clique. Moreover, we show the monotonicity of the eigenvalues when passing to spanning subgraphs and the monotonicity of magnetic Cheeger constants with respect to the geometric preorder. Finally, we prove a refined procedure to detect spectral gaps in the spectrum of an infinite covering graph.
In this paper, new concepts of covering of fuzzy graphs are introduced. The definitions of fuzzy covering maps and fuzzy covering graphs of fuzzy graphs are given. Some special types of fuzzy ...covering graphs are discussed. Some important theorems to find out the fuzzy covering maps as well as fuzzy covering graphs for different types of fuzzy graphs are described. On the basis of the present situation during the current pandemic COVID-19, the world economic status is highly disruptive and deeply affected by the lock-down process. So, this topic is catching the eye for being an application part of this paper. Also, there are globally seventeen goals to sustain our development, which contain the eighth goal ‘Decent Work and Economic Growth’ having effect on the economy of the world. For this reason, the eighth sustainable development goal is combined with the economic impact of the pandemic for the discussion in the application part. Then some strategies are made to overcome this condition in a better way and perform all the steps to get a better situation in near future.
In this article, we analyze the spectrum of discrete magnetic Laplacians (DML) on an infinite covering graph G ˜ → G = G ˜ / Γ with (Abelian) lattice group Γ and periodic magnetic potential β ˜ . We ...give sufficient conditions for the existence of spectral gaps in the spectrum of the DML and study how these depend on β ˜ . The magnetic potential can be interpreted as a control parameter for the spectral bands and gaps. We apply these results to describe the spectral band/gap structure of polymers (polyacetylene) and nanoribbons in the presence of a constant magnetic field.
We give a new construction of Ramanujan graphs using a generalized type of covering graph called a weighted covering graph. For a given prime p the basic construction produces bipartite Ramanujan ...graphs with 4p vertices and degrees 2N where roughly $p + 1 - \sqrt{2p} < N \le p$ . We then give generalizations to produce Ramanujan graphs of other sizes and degrees as well as general results about base graphs which have weighted covers that satisfy their Ramanujan bounds. To do the construction, we define weighted covering graphs and distinguish a subclass of Galois weighted covers that allows for block diagonalization of the adjacency matrix. The specific construction allows for easy computation of the resulting blocks. The Gershgorin Circle Theorem is then used to compute the Ramanujan bounds on the spectra.
We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using various types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the ...equilateral metric and discrete graph spectrum (also in the “exceptional” values of the metric graph corresponding to the Dirichlet spectrum) we carry over these estimates from the metric graph Laplacian to the discrete case. We apply the results to covering graphs and present examples where the covering graph Laplacians have spectral gaps.
The
clique graph
kG of a graph G is the intersection graph of the family of all maximal complete subgraphs of G. The iterated clique graphs
k
n
G are defined by
k
0
G=
G and
k
n+1
G=
kk
n
G. A graph ...G is said to be k-
divergent if
V(k
nG)
tends to infinity with n. A graph is
locally
C
6 if the neighbours of any given vertex induce an hexagon. We prove that all locally
C
6 graphs are k-divergent and that the diameters of the iterated clique graphs also tend to infinity with n while the sizes of the cliques remain bounded.
We apply two methods to the block diagonalization of the adjacency matrix of the Cayley graph defined on the affine group. The affine group will be defined over the finite ring
Z
/
p
n
Z
. The method ...of irreducible representations will allow us to find nontrivial eigenvalue bounds for two different graphs. One of these bounds will result in a family of Ramanujan graphs. The method of covering graphs will be used to block diagonalize the affine graphs using a Galois group of graph automorphisms. In addition, we will demonstrate the method of covering graphs on a generalized version of the graphs of Lubotzky et al. A. Lubotzky, R. Phillips, P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988) 261–277.
This paper presents a synthesis algorithm, Covering Set Partitions (CSP), for reversible binary functions with no ancillary (garbage) bits. Existing algorithms are constrained to functions of small ...number of variables because they store the entire truth table of 2n terms in memory or require a huge amount of time to yield results because they must calculate all possible permutations of an input vector. In contrast, the CSP algorithm harnesses the natural mathematical properties of binary numbers, partially ordered sets and covering graph theory, to construct input vectors which are guaranteed to produce valid results. A randomly selected subset of all valid input vectors are processed where the best input vector sequence wins. The CSP algorithm is capable of synthesizing functions of large number of variables (30 bits) in a reasonable amount of time.
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