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This is an example of a regular of degree 3. Each node has 3 connections in the graph. There are several properties concerning the eigenvalues of regular graphs. Regular graph spectrum. Definition: If the simple graph G is m-regular, then (1) m is an eigenvalue of G (2) the multiplicity of the eigenvalue m is 1, provided G is connected (3 ...

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A natural special case of this process is the Erd}os-R enyi graph model G(n;p) which is the special case where the host graph is Kn. Other examples are the percolation problems that have long been studied [10,11] in theoretical physics, mainly with the host graph being the lattice graph Zk. In this paper, we consider a general host graph, an ...

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Note that for any graph, the adjacency matrix is real symmetric. For d-regular graphs, each row and column has precisely d 1's (and all the other entries are 0). Henceforth, if we refer to the eigenvalues of a graph, we really mean the eigenvalues of the associated adjacency matrix. 1.1 Weighted Graphs

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host graph is d-regular with adjacency eigenvalue and they show that the critical probability is close to 1=d, strengthening earlier results on hypercubes [2] and Cayley graphs [12]. For expander graphs with degrees bounded by d, Alon, Benjamini and Stacey [1] proved that the percolation threshold is greater than or equal to 1=(2d).

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Sep 30, 2016 · Repeat for \(k\) steps or until convergence. In practice, the Weisfeiler-Lehman algorithm assigns a unique set of features for most graphs. This means that every node is assigned a feature that uniquely describes its role in the graph. Exceptions are highly regular graphs like grids, chains, etc.

Randerath extended this to l-edge-connected k-regular graphs. • How small can the matching number of an n-vertex connected cubic graph be? The answer was determined by Biedl et al. Broere et al. gave a formula for the minimum size of a matching in a (k −2)-edge-connected k-regular graph with a ﬁxed number of vertices.

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be eigenvalues of G. Now it follows that G cannot have precisely two distinct degrees, if G has nonintegral eigenvalues. In case G has nonintegral eigenvalues, we moreover have the following proposition, which generalizes the ''conference case'' (''half-case'') for strongly regular graphs. NOTE 195

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eigenvalue of the transition matrix. In this paper we address the problem of assigning probabilities to the edges of the graph in such a way as to minimize the second largest magnitude eigenvalue, i.e., the problem of ﬂnding the fastest mixing Markov chain on the graph. We show that this problem can be formulated as a convex optimization ...

4. Show that a connected d-regular graph is bipartite iﬀ the least eigenvalue of its adjacency matrix is −d. Solution: 4 5. Compute the eigenvalues and eigenvectors of the following graphs: (a) Kn, the complete graph on n vertices. (b) Kn,n, the complete bipartite graph with partites of size n each. (c) Cn, the cycle on n vertices. Solution: 5

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2. 2 2. 3. k= 0 i Ghas at least kconnected components.(In particular, 2 >0 i Gis connected.) 4. n= 2 i at least one connected component is bipartite. Proof: Note that if x2Rn, then xTLx= 1 d P (u;v)2E (x u x v) 2 and also 1 = min x2Rn˜f0g xTLx xTx 0: Take ~1 = (1 ;:::;1), in which case ~1TL~1 = 0, and so 0 is the smallest eigenvalue, and ~1 is one of the eigenvectors in the eigenspace of this ...

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graph consisting of the n2 nodes, f(i;j)j1 i;j ng. Then edges connect any two nodes that agree in one coordinate, and hence the graph is regular of degree 2n 2. Figure 1. Legal Moves on the Rook Graph The transition matrix for the random walk on the rook graph is given by P (i;j)(k;l) = (1 2n if i =kor j l 0 else :

On asymmetric colourings of graphs with bounded degrees and infinite motion. with M. Pilśniak and M. Stawiski 2019, preprint. pdf [29] A bound for the distinguishing index of regular graphs. with M. Pilśniak and M. Stawiski European Journal of Combinatorics, 89, 2020. doi pdf [28] Bounding the cop number of a graph by its genus.

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connectedness of the graph. First we prove that a graph has k connected components if and only if the algebraic multiplicity of eigenvalue 0 for the graph’s Laplacian matrix is k. We then prove Cheeger’s inequality (for d-regular graphs) which bounds the number of edges between the two subgraphs

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k 0 z khe u;A ke vi: Observe also that he v;R(z)e vi= Z d ev G (x) x z (4) is the Cauchy-Stieltjes transform of ev G. In these notes, we will mostly be interested by the regularity properties of the measure ev G. For some explicit computation of spectral measures in regular graphs, see examples below, Hora and Obata [49] and for a recent ...

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connected. The . distance. between any two vertices in a graph is the number of edges "traced" on the shortest walk between the two vertices. The distance from . a. to . d. is 2. The maximum distance between all of the pairs of vertices of a graph is the . diameter. of the graph. In Figure 1-1, the distance between any two vertices is ...Feb 17, 2012 · (2016) A characterization of strongly regular graphs in terms of the largest signless Laplacian eigenvalues. Linear Algebra and its Applications 506 , 1-5. (2016) Some results on the Laplacian spread of a graph.

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Corollary 1.1. The complete graph K nis the only graph on nvertices with eigenvalue n 1. In particular, K n is uniquely determined by its spectrum. Remark. Not all graphs can be recovered by their spectrum, i.e., the spectrum is not a faithful (in that graph can be uniquely determined) graph invariant (in that spectrum depends on

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A connected regular graph with at most three distinct eigenvalues is known to be strongly regular, and see, for example [3] for a survey on strongly regular graphs. The connected non-regular ...

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Graphs of polyhedra; polyhedra as graphs. Abstract Relations between graph theory and polyhedra are presented in two contexts. In the first, the symbiotic dependence between 3-connected planar graphs and convex polyhedra is described in detail. In the second, a theory of nonconvex polyhedra is based on a graph-theoretic foundation.

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