The number of connected graphs on n labeled points and q lines (no loops, no multiple lines) is f(n,q). In the first paper of this series I showed how to find an (increasingly complicated) exact formula for f(n,n+k) for general n and successive k. The method would give an ...
摘要: For the independence number α( G) of a connected graph G on n vertices with m edges the inequality α(G)⩾ 1 2[(2m+n+1)− (2m+n+1) 2−4n 2] is proved and its algorithmic realization is discussed.关键词:Independence Stability ...
Bermond proved that a 2-connected graph on n vertices always contains a path (a 2-leaf subtree) of length min {n,δ 2 }, where δ 2 is the minimum degree-sum of a 2-element independent set. We generalize this result to obtain a sufficient condition for the existence of a large q ...
For all n, c, we determine the maximum number of connected induced subgraphs in a connected graph with order n and c cut vertices, and also characterise those graphs attaining the bound. Moreover, we show that the cycle has the smallest number of connected induced subgraphs among all cut ...
The article discusses the study on the independence number of distance graphs with vertices in {1, 0, 1} n wherein only one linear-algebraic method was known to be effective in obtaining the upper bounds of vertices. It notes that the authors have developed a technique...
time algorithm for deciding the divergence or convergence of a graph. In this work we prove that any graph with at least 7 bicliques diverges under the biclique operator. Furthermore, we prove that graphs with no twin vertices that are not divergent have at most 12 vertices, which leads to...
It is sufficient to prove that cr(G) ≤ cd(n − 1) holds for any two-connected graph G satisfying the conditions. Indeed, if G is discon- nected or has a cut vertex, then it can be obtained as the union of two graphs G1 and G2 with n1 and n2 vertices that have at most ...
The hyper-Wiener index of graphs with a given chromatic (clique) number Let G be a simple connected graph. The Wiener index W(G) is the sum of all distances between vertices of G, whereas the hyper-Wiener index WW(G) is defined... L Feng,G Yu,W Liu - 《Utilitas Mathematica》 被...
They also give a few examples of graphs for which the eternal vertex cover number and the vertex cover number coincide; such as the complete graph on n vertices (Kn), the Petersen graph, , (where represents the box product) and the n×m grid, where either n or m is even. However, ...
Finding Independent Sets inK4Free 4Regular Connected Graphs Let G be a K4-free 4-regular connected graph with nuvertices. Then G contains an independent set of vertices ofcardinality at least (7nu-4)/26. Furthermore, the proofwill yield a polynomial-time algorithm which will return anindepend...