A set of vertices in a graph is perfect dominating if every vertex outside the set is adjacent to exactly one vertex in the set, and is neighborhood connected if the subgraph induced by its open neighborhood is connected. In any graph the full set of vertices is perfect dominating, and in every connected graph the full set of vertices is neighborhood connected. It is shown that (i) in a connect...

In this note we address the problem of graph isomorphism by means of eigenvalue spectra of different matrix representations: the neighborhood matrix M̂ , its corresponding signless Laplacian QM̂ , and the set of higher order adjacency matrices M`s. We find that, in relation to graphs with at most 10 vertices, QM̂ leads to better results than the signless Laplacian Q; besides, when combined with M̂ ...

a graph $g$ is called a fractional‎ ‎$(k,n',m)$-critical deleted graph if any $n'$ vertices are removed‎ ‎from $g$ the resulting graph is a fractional $(k,m)$-deleted‎ ‎graph‎. ‎in this paper‎, ‎we prove that for integers $kge 2$‎, ‎$n',mge0$‎, ‎$nge8k+n'+4m-7$‎, ‎and $delta(g)ge k+n'+m$‎, ‎if‎ ‎$$|n_{g}(x)cup n_{g}(y)|gefrac{n+n'}{2}$$‎ ‎for each pair of non-adjac...

We consider simulations of graph automata. We introduce two local transformations on the neighborhood: splitting and merging. We explain how to use such transformations, and their consequences on the topology of the simulated graph, the speed of the simulation and the memory size of simulating automata in some cases. As an example , we apply these transformations to graph automata embedded on s...

`-Graph, which learns a sparse graph over the data by sparse representation, has been demonstrated to be effective in clustering especially for high dimensional data. Although it achieves compelling performance, the sparse graph generated by `-Graph ignores the geometric information of the data by sparse representation for each datum separately. To obtain a sparse graph that is aligned to the u...

Let $G$ be a graph with vertex set $V(G)$. For any integer $kge 1$, a signed (total) $k$-dominating functionis a function $f: V(G) rightarrow { -1, 1}$ satisfying $sum_{xin N[v]}f(x)ge k$ ($sum_{xin N(v)}f(x)ge k$)for every $vin V(G)$, where $N(v)$ is the neighborhood of $v$ and $N[v]=N(v)cup{v}$. The minimum of the values$sum_{vin V(G)}f(v)$, taken over all signed (total) $k$-dominating functi...

Spectral manifold learning techniques have recently found extensive applications in machine vision. The common strategy of spectral algorithms for manifold learning is exploiting the local relationships in a symmetric adjacency graph, which is typically constructed using -nearest neighborhood ( -NN) criterion. In this paper, with our focus on locally linear embedding as a powerful and well-know...

We introduce a novel graph kernel called the Neighborhood Subgraph Pairwise Distance Kernel. The kernel decomposes a graph into all pairs of neighborhood subgraphs of small radius at increasing distances. We show that using a fast graph invariant we obtain significant speed-ups in the Gram matrix computation. Finally, we test the novel kernel on a wide range of chemoinformatics tasks, from anti...

Quality of collective inference relational graph classi er depends on a degree of homophily in a classi ed graph. If we increase homophily in the graph, the classi er would assign class-membership to the instances with reduced error rate. We propose to substitute traditionally used graph neighborhood method (based on direct neighborhood of vertex) with local graph ranking algorithm (activation ...

We introduce the adaptive neighborhood graph as a data structure for modeling a smooth manifold M embedded in some Euclidean space . We assume that M is known to us only through a finite sample P ⊂ M , as it is often the case in applications. The adaptive neighborhood graph is a geometric graph on P . Its complexity is at most min{2O(k)n, n2}, where n = |P | and k = dim M , as opposed to the n ...