Dans cet article on étudie les propriétés d’ordres totaux à distance minimum d’un ensemble de tournois ; on montre, par exemple, que ces ordres contiennent l’ordre d’unanimité. On étudie la fonction $f(n,v)$ maximum de la distance entre un ordre total et $v$ tournois définis sur un ensemble à $n$ éléments ; on donne sa valeur exacte pour $v$ pair, un encadrement pour $v$ impair, et sa valeur limite pour $v$ tendant vers l’infini.

We investigate which switching classes do not contain a bipartite graph. Our final aim is a characterization by means of a set of critically non-bipartite graphs: they do not have a bipartite switch, but every induced proper subgraph does. In addition to the odd cycles, we list a number of exceptional cases and prove that these are indeed critically non-bipartite. Finally, we give a number of structural results towards proving the fact that we have indeed found them all. The search for critically...

In [Mwambene E., Multiples of left loops and vertex-transitive graphs, Cent. Eur. J. Math. 3 (2005), no. 2, 254–250] it was proved that every vertex-transitive graph is the Cayley graph of a left loop with respect to a quasi-associative Cayley set. We use this result to show that Cayley graphs of left loops with respect to such sets have some properties in common with Cayley graphs of groups which can be used to study a geometric theory for left loops in analogy to that for groups.

In this work, Transition Probability Matrix (TPM) is proposed as a new method for extracting the features of nodes in the graph. The proposed method uses random walks to capture the connectivity structure of a node's close neighborhood. The information obtained from random walks is converted to anonymous walks to extract the topological features of nodes. In the embedding process of nodes, anonymous walks are used since they capture the topological similarities of connectivities better than random...

A graph $G$ is called $\{H_1,H_2,\cdots ,H_k\}$-free if $G$ contains no induced subgraph isomorphic to any graph $H_i$, $1\le i\le k$. We define $${\sigma }_k=min\left\{\sum _{i=1}^{k}d\left(v_i\right):\{v_1,\cdots ,v_k\}\text{is}\text{an}\text{independent}\text{set}\text{of}\text{vertices}\text{in}G\right\}.$$ In this paper, we prove that (1) if $G$ is a connected $\{K_{1,4},K_{1,4}+e\}$-free graph of order $n$ and ${\sigma }_3\left(G\right)\ge n-1$, then $G$ is traceable, (2) if $G$ is a 2-connected $\{K_{1,4},K_{1,4}+e\}$-free graph of order $n$ and $|N\left(x_1\right)\cup N\left(x_2\right)|+|N\left(y_1\right)\cup N\left(y_2\right)|\ge n-1$ for any two distinct pairs of non-adjacent vertices $\{x_1,x_2\}$, $\{y_1,y_2\}$ of $G$, then $G$ is traceable, i.e., $G$ has a Hamilton path, where $K_{1,4}+e$ is a graph obtained by joining a pair of non-adjacent vertices in a $K_{1,4}$.

First, noncompact Cantor sets along with their defining trees are introduced as a natural generalization of $p$-adic numbers. Secondly we construct a class of jump processes on a noncompact Cantor set from given pairs of eigenvalues and measures. At the same time, we have concrete expressions of the associated jump kernels and transition densities. Then we construct intrinsic metrics on noncompact Cantor set to obtain estimates of transition densities and jump kernels under some regularity conditions...