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Register a new userOn Resolvability of a Graph Associated to a Finite Vector Space
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The metric dimension of non-component graph, associated to a finite vector space, is determined. It is proved that the exchange property holds for resolving sets of the graph, except a special case. Some results are also related to an intersection graph.
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In this paper, we investigate the problem of covering the vertices of a graph associated to a finite vector space as introduced by Das cite{Das}, such that we can uniquely identify any vertex by examining the vertices that cover it. We use locating-dominating sets and identifying codes, which are closely related concepts for this purpose. These sets consist of a dominating set of graph such that every vertex is uniquely identified by its neighborhood within the dominating sets. We find the location-domination number and the identifying number of the graph and study the exchange property for locating-dominating sets and identifying codes.
Let $mathbb{E}(H)$ and $mathbb{V}(H)$ denote the edge set and the vertex set of the simple connected graph $H$, respectively. The mixed metric dimension of the graph $H$ is the graph invariant, which is the mixture of two important graph parameters, the edge metric dimension and the metric dimension. In this article, we compute the mixed metric dimension for the two families of the plane graphs viz., the Web graph $mathbb{W}_{n}$ and the Prism allied graph $mathbb{D}_{n}^{t}$. We show that the mixed metric dimension is non-constant unbounded for these two families of the plane graph. Moreover, for the Web graph $mathbb{W}_{n}$ and the Prism allied graph $mathbb{D}_{n}^{t}$, we unveil that the mixed metric basis set $M_{G}^{m}$ is independent.
Write $mathcal{C}(G)$ for the cycle space of a graph $G$, $mathcal{C}_kappa(G)$ for the subspace of $mathcal{C}(G)$ spanned by the copies of the $kappa$-cycle $C_kappa$ in $G$, $mathcal{T}_kappa$ for the class of graphs satisfying $mathcal{C}_kappa(G)=mathcal{C}(G)$, and $mathcal{Q}_kappa$ for the class of graphs each of whose edges lies in a $C_kappa$. We prove that for every odd $kappa geq 3$ and $G=G_{n,p}$, [max_p , Pr(G in mathcal{Q}_kappa setminus mathcal{T}_kappa) rightarrow 0;] so the $C_kappa$s of a random graph span its cycle space as soon as they cover its edges. For $kappa=3$ this was shown by DeMarco, Hamm and Kahn (2013).
We study perfect state transfer in Grover walks, which are typical discrete-time quantum walk models. In particular, we focus on states associated to vertices of a graph. We call such states vertex type states. Perfect state transfer between vertex type states can be studied via Chebyshev polynomials. We derive a necessary condition on eigenvalues of a graph for perfect state transfer between vertex type states to occur. In addition, we perfectly determine the complete multipartite graphs whose partite sets are the same size on which perfect state transfer occurs between vertex type states, together with the time.
For a graph whose vertex set is a finite set of points in $mathbb R^d$, consider the closed (open) balls with diameters induced by its edges. The graph is called a (an open) Tverberg graph if these closed (open) balls intersect. Using the idea of halving lines, we show that (i) for any finite set of points in the plane, there exists a Hamiltonian cycle that is a Tverberg graph; (ii) for any $n$ red and $n$ blue points in the plane, there exists a perfect red-blue matching that is a Tverberg graph. Using the idea of infinite descent, we prove that (iii) for any even set of points in $mathbb R^d$, there exists a perfect matching that is an open Tverberg graph; (iv) for any $n$ red and $n$ blue points in $ mathbb R^d $, there exists a perfect red-blue matching that is a Tverberg graph.
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