Identifying the complementary relationship between products, like a cartridge to a printer, is a very useful technique to provide recommendations. These are typically purchased together or within a short time frame and thus online retailers benefit from it. Existing approaches rely heavily on transactions and therefore they suffer from: (1) the cold start problem for new products; (2) the inabi...

A/Mmcr-It is proved that the Shannon zero-error capacity of the pentagon is e. The method is then generalized to obtain upper bounds on the capacity of au arbitrary graph. A well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases. Several results are obtained on the capacity of special g...

One of the basic facts of group theory is that each finite group contains a Sylow p-subgroup for each prime p which divides the order of the group. In this note we show that each vertex-transitive selfcomplementary graph has an analogous property. As a consequence of this fact, we obtain that each prime divisor p of the order of a vertex-transitive self-complementary graph satisfies the congrue...

As the central notion in semi-supervised learning, smoothness is often realized on a graph representation of the data. In this paper, we study two complementary dimensions of smoothness: its pointwise nature and probabilistic modeling. While no existing graph-based work exploits them in conjunction, we encompass both in a novel framework of Probabilistic Graph-based Pointwise Smoothness (PGP), ...

By a graph X we mean a finite set V(X), called the vertices of X, together with a set E(X), called the edges of X, consisting of unordered pairs of distinct elements of V(X). We shall indicate the unordered pairs by brackets. Two graphs X and Y are said to be isomorphic, denoted by X cz Y, if there is a one-to-one map a of V(X) onto V(Y) such that [aa,ab~\eE(Y) if and only if [a, b~]eE(X). An i...

The complementary prism GG of a graph G is formed from the disjoint union of G and its complement G by adding the edges of a perfect matching between the corresponding vertices of G and G. A Roman dominating function on a graph G = (V,E) is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 is adjacent to a vertex with label 2. The Roman domination number γR(G) ofG is the mini...

Abstract We present a memory-augmented approach to condition an autoregressive language model on knowledge graph. represent the graph as collection of relation triples and retrieve relevant relations for given context improve text generation. Experiments WikiText-103, WMT19, enwik8 English datasets demonstrate that our produces better in terms perplexity bits per character. also show relational...

In the geodetic convexity, a set of vertices S graph G is convex if all belonging to any shortest path between two lie in . The hull H ( ) smallest containing If = V ), then cardinality h minimum number complementary prism GḠ arises from disjoint union and Ḡ by adding edges perfect matching corresponding A autoconnected both are connected. Motivated previous work, we study for prisms graphs. Wh...

Social network is a graph G = [V,E], in which each vertex from V corresponds to a person and each edge from E describes, that two persons know each other. Social substructure of size m is defined as a clique Km in social network G with m of its vertices. In other words, in a social substructure, everybody knows everybody. In this paper, we study the complementary graph G, in which edges represe...

In a series of four papers we determine structures whose existence is dual, in the sense complementary, to stars or combs from well-known star—comb lemma for infinite graphs. Call set U vertices graph G tough if only finitely many components ? X meet every finite vertex ? V ( ) . this fourth and final paper series, structurally characterise connected graphs which given tough. Our characterisati...