Two-point resistances and random walks on stellated regular graphs

Yang, Yujun; Klein, Douglas J.

doi:10.1088/1751-8121/aaf8e7

Cited by 15 publications

(6 citation statements)

References 47 publications

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“…See [7,16,12,21,23,22] for some recent researches on effective resistance and Kirchhoff index. Now we introduce a nice relation between electrical network and spanning trees of graphs.…”

Section: Electrical Networkmentioning

confidence: 99%

Spanning trees in complete bipartite graphs and resistance distance in nearly complete bipartite graphs

Dong

2020

Discrete Applied Mathematics

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Using the theory of electrical network, we first obtain a simple formula for the number of spanning trees of a complete bipartite graph containing a certain matching or a certain tree.Then we apply the effective resistance (i.e., resistance distance in graphs) to find a formula for the number of spanning trees in the nearly complete bipartite graph G(m, n, p) = Km,n − pK2 (p ≤ min{m, n}), which extends a recent result by Ye and Yan who obtained the effective resistances and the number of spanning trees in G(n, n, p). As a corollary, we obtain the Kirchhoff index of G(m, n, p) which extends a previous result by Shi and Chen.

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“…See [7,16,12,21,23,22] for some recent researches on effective resistance and Kirchhoff index. Now we introduce a nice relation between electrical network and spanning trees of graphs.…”

Section: Electrical Networkmentioning

confidence: 99%

Spanning trees in complete bipartite graphs and resistance distance in nearly complete bipartite graphs

Dong

2020

Discrete Applied Mathematics

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“…In [31], Shangguan and Chen obtained resistance distances in the vertex-face graph of a planar graph. In [32], resistance distances and Kirchhoff indices of stellated graphs were obtained.…”

Section: Introductionmentioning

confidence: 99%

Resistance Distances and Kirchhoff Indices Under Graph Operations

Yang

2020

IEEE Access

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The resistance distance between any two vertices of a connected graph G is defined as the net effective resistance between them in the electrical network constructed from G by replacing each edge with a unit resistor. The Kirchhoff index of G is defined as the sum of resistance distances between all pairs of vertices. In this paper, two unary graph operations on G are taken into consideration, with the resulted graphs being denoted by RT (G) and H (G). Using electrical network approach and combinatorial approach, we derive explicit formulae for resistance distances and Kirchhoff indices of RT (G) and H (G). It turns out that resistance distances and Kirchhoff indices of RT (G) and H (G) could be expressed in terms of resistance distances and graph invariants of G. Our result generalizes the previously known result on the Kirchhoff index of RT (G) for a regular graph G to the Kirchhoff index of RT (G) for an arbitrary graph G.INDEX TERMS Resistance distance, Kirchhoff index, multiplicative degree-Kirchhoff index, additive degree-Kirchhoff index, Foster's formula.

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“…The distance between two leaves of the network is found by considering the edge weights as electrical resistance, obeying Ohm's law. The metric resistance distance for all nodes (not only leaves) of a graph is introduced in Klein and Randić (1993), and studied closely in subsequent papers such as Yang and Klein (2015) and Yang and Klein (2019). To study graphs, the resistance of each edge is often assumed to have unit value, but the definitions allow any weight.…”

Section: Introductionmentioning

confidence: 99%

Phylogenetic Networks as Circuits With Resistance Distance

Forcey

Scalzo

2020

Front. Genet.

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Phylogenetic networks are notoriously difficult to reconstruct. Here we suggest that it can be useful to view unknown genetic distance along edges in phylogenetic networks as analogous to unknown resistance in electric circuits. This resistance distance, well-known in graph theory, turns out to have nice mathematical properties which allow the precise reconstruction of networks. Specifically we show that the resistance distance for a weighted 1-nested network is Kalmanson, and that the unique associated circular split network fully represents the splits of the original phylogenetic network (or circuit). In fact, this full representation corresponds to a face of the balanced minimal evolution polytope for level-1 networks. Thus, the unweighted class of the original network can be reconstructed by either the greedy algorithm neighbor-net or by linear programming over a balanced minimal evolution polytope. We begin study of 2-nested networks with both minimum path and resistance distance, and include some counting results for 2-nested networks.

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Two-point resistances and random walks on stellated regular graphs

Cited by 15 publications

References 47 publications

Spanning trees in complete bipartite graphs and resistance distance in nearly complete bipartite graphs

Spanning trees in complete bipartite graphs and resistance distance in nearly complete bipartite graphs

Resistance Distances and Kirchhoff Indices Under Graph Operations

Phylogenetic Networks as Circuits With Resistance Distance

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