The signless Laplacian spectral radius of bicyclic graphs with a given girth.
The spectral method for detrmining the number of trees.
The spectrum of the edge corona of two graphs.
The star is the tree with greatest greatest Laplacian eigenvalue
The Steiner Minimal Network for Convex Configurations.
The Steiner Ratio Conjecture for Cocircular Points.
The Steiner Wiener Index of A Graph
The Wiener index W(G) of a connected graph G, introduced by Wiener in 1947, is defined as W(G) = ∑u,v∈V(G) d(u, v) where dG(u, v) is the distance between vertices u and v of G. The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph G of order at least 2 and S ⊆ V (G), the Steiner distance d(S) of the vertices of S is the minimum size of a connected subgraph whose vertex set is S. We now...
The structure of locally finite two-connected graphs.
The tree of shapes of an image
In [30], Kronrod proves that the connected components of isolevel sets of a continuous function can be endowed with a tree structure. Obviously, the connected components of upper level sets are an inclusion tree, and the same is true for connected components of lower level sets. We prove that in the case of semicontinuous functions, those trees can be merged into a single one, which, following its use in image processing, we call “tree of shapes”. This permits us to solve a classical representation...
The tree of shapes of an image
In [CITE], Kronrod proves that the connected components of isolevel sets of a continuous function can be endowed with a tree structure. Obviously, the connected components of upper level sets are an inclusion tree, and the same is true for connected components of lower level sets. We prove that in the case of semicontinuous functions, those trees can be merged into a single one, which, following its use in image processing, we call “tree of shapes”. This permits us to solve a classical representation problem...
The tree-covering number of a graph
The triangles method to build -trees from incomplete distance matrices
A method to infer -trees (valued trees having as set of leaves) from incomplete distance arrays (where some entries are uncertain or unknown) is described. It allows us to build an unrooted tree using only 2-3 distance values between the elements of , if they fulfill some explicit conditions. This construction is based on the mapping between -tree and a weighted generalized 2-tree spanning .
The triangles method to build X-trees from incomplete distance matrices
A method to infer X-trees (valued trees having X as set of leaves) from incomplete distance arrays (where some entries are uncertain or unknown) is described. It allows us to build an unrooted tree using only 2n-3 distance values between the n elements of X, if they fulfill some explicit conditions. This construction is based on the mapping between X-tree and a weighted generalized 2-tree spanning X.
The Turán Number of the Graph 2P5
We give the Turán number ex (n, 2P5) for all positive integers n, improving one of the results of Bushaw and Kettle [Turán numbers of multiple paths and equibipartite forests, Combininatorics, Probability and Computing, 20 (2011) 837-853]. In particular we prove that ex (n, 2P5) = 3n−5 for n ≥ 18.
The Turàn number of the graph 3P4
Let ex (n,G) denote the maximum number of edges in a graph on n vertices which does not contain G as a subgraph. Let Pi denote a path consisting of i vertices and let mPi denote m disjoint copies of Pi. In this paper we count ex(n, 3P4)
The Tutte polynomial of a graph, depth-first search, and simplicial complex partitions.
The Vertex-Rainbow Index of A Graph
The k-rainbow index rxk(G) of a connected graph G was introduced by Chartrand, Okamoto and Zhang in 2010. As a natural counterpart of the k-rainbow index, we introduce the concept of k-vertex-rainbow index rvxk(G) in this paper. In this paper, sharp upper and lower bounds of rvxk(G) are given for a connected graph G of order n, that is, 0 ≤ rvxk(G) ≤ n − 2. We obtain Nordhaus-Gaddum results for 3-vertex-rainbow index of a graph G of order n, and show that rvx3(G) + rvx3(Ḡ) = 4 for n = 4 and 2 ≤...
Théorie des magmoïdes
Théorie des magmoïdes (I)
Three new heuristics for the Steiner problem in graphs.