Barycentric subdivision

2015 ◽  
pp. 185-202
Author(s):  
T. Kyle Petersen
Keyword(s):  
Barycentric Subdivision

scholarly journals The insufficiency of barycentric subdivision.

1965 ◽  
Vol 12 (3) ◽  
pp. 263-272 ◽  
Author(s):  
Ross L. Finney
Keyword(s):  
Barycentric Subdivision

scholarly journals On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs

2016 ◽  
Vol 14 (1) ◽  
pp. 641-648 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Spanning Trees ◽  
Line Graph ◽  
Upper And Lower Bounds ◽  
Barycentric Subdivision ◽  
Electrical Networks ◽  
Zagreb Index ◽  
Laplacian Eigenvalues ◽  
Estrada Index ◽  
Vertex Degrees ◽  
Number Of Spanning Trees

Abstract As a generalization of the Sierpiński-like graphs, the subdivided-line graph Г(G) of a simple connected graph G is defined to be the line graph of the barycentric subdivision of G. In this paper we obtain a closed-form formula for the enumeration of spanning trees in Г(G), employing the theory of electrical networks. We present bounds for the largest and second smallest Laplacian eigenvalues of Г(G) in terms of the maximum degree, the number of edges, and the first Zagreb index of G. In addition, we establish upper and lower bounds for the Laplacian Estrada index of Г(G) based on the vertex degrees of G. These bounds are also connected with the number of spanning trees in Г(G).

scholarly journals BOND PERCOLATION ON A NON-P.C.F. SIERPIŃSKI GASKET, ITERATED BARYCENTRIC SUBDIVISION OF A TRIANGLE, AND HEXACARPET

Fractals ◽  
2016 ◽  
Vol 24 (02) ◽  
pp. 1650023
Author(s):  
D. LOUGEE ◽  
B. STEINHURST
Keyword(s):  
Phase Transition ◽  
Sierpinski Gasket ◽  
Barycentric Subdivision ◽  
Bond Percolation ◽  
Critical Probability ◽  
Sierpiński Gasket ◽  
Dual Graphs

We investigate bond percolation on the iterated barycentric subdivision of a triangle, the hexa-carpet, and the non-p.c.f. Sierpinski gasket. With the use of known results on the diamond fractal, we are able to bound the critical probability of bond percolation on the non-p.c.f. gasket and the iterated barycentric subdivision of a triangle from above by 0.282. We then show how both the gasket and hexacarpet fractals are related via the iterated barycentric subdivisions of a triangle: the two spaces exhibit duality properties although they are not themselves dual graphs. Finally, we show the existence of a non-trivial phase transition on all three graphs.

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