The Categorical Product of Graphs

1968 ◽  
Vol 20 ◽  
pp. 1511-1521 ◽  
Author(s):  
Donald J. Miller
Keyword(s):  
Kronecker Product ◽  
Undirected Graphs ◽  
Graph Homomorphisms ◽  
Product Of Graphs

Undirected graphs and graph homomorphisms as introduced by Sabidussi (6, p. 386), form a category that admits a categorical product. For the category of graphs and full graph homomorphisms, the categorical product was introduced by Čulik (1) under the name cardinal product. It was independently defined by Weichsel (8) who called it the Kronecker product and investigated the connectedness of products of finitely many factors. Hedetniemi (4) was the first to make use of the fact that the cardinal product is categorical.

Kronecker Products and Local Joins of Graphs

1977 ◽  
Vol 29 (2) ◽  
pp. 255-269 ◽  
Author(s):  
M. Farzan ◽  
D. A. Waller
Keyword(s):  
Tensor Product ◽  
Kronecker Product ◽  
Kronecker Products ◽  
Finite Graphs ◽  
Product Of Graphs

When studying the category raph of finite graphs and their morphisms, Ave can exploit the fact that this category has products, [we define these ideas in detail in § 2]. This categorical product of graphs is usually called their Kronecker product, though it has been approached by various authors in various ways and under various names, including tensor product, cardinal product, conjunction and of course categorical product (see for example [6; 7; 11 ; 14; 17 and 23]).

scholarly journals Proof of a conjecture on connectivity of Kronecker product of graphs

2011 ◽  
Vol 311 (21) ◽  
pp. 2563-2565 ◽  
Author(s):  
Yun Wang ◽  
Baoyindureng Wu
Keyword(s):  
Kronecker Product ◽  
Product Of Graphs

Some remarks on the Kronecker product of graphs

1998 ◽  
Vol 68 (2) ◽  
pp. 55-61 ◽  
Author(s):  
Anne Bottreau ◽  
Yves Métivier
Keyword(s):  
Kronecker Product ◽  
Product Of Graphs

scholarly journals Shannon Capacity and the Categorical Product

10.37236/9113 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Gábor Simonyi
Keyword(s):  
Lower Bound ◽  
Recent Result ◽  
Chromatic Number ◽  
Upper Bound ◽  
Graph Invariant ◽  
Shannon Capacity ◽  
Graph Homomorphisms ◽  
Complementary Graph ◽  
Product Of Graphs ◽  
Monotone Graph

Shannon OR-capacity $C_{\rm OR}(G)$ of a graph $G$, that is the traditionally more often used Shannon AND-capacity of the complementary graph, is a homomorphism monotone graph parameter therefore $C_{\rm OR}(F\times G)\leqslant\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$ holds for every pair of graphs, where $F\times G$ is the categorical product of graphs $F$ and $G$. Here we initiate the study of the question when could we expect equality in this inequality. Using a strong recent result of Zuiddam, we show that if this "Hedetniemi-type" equality is not satisfied for some pair of graphs then the analogous equality is also not satisfied for this graph pair by some other graph invariant that has a much "nicer" behavior concerning some different graph operations. In particular, unlike Shannon OR-capacity or the chromatic number, this other invariant is both multiplicative under the OR-product and additive under the join operation, while it is also nondecreasing along graph homomorphisms. We also present a natural lower bound on $C_{\rm OR}(F\times G)$ and elaborate on the question of how to find graph pairs for which it is known to be strictly less than the upper bound $\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$. We present such graph pairs using the properties of Paley graphs.

The Restricted Edge-Connectivity of Kronecker Product Graphs

2019 ◽  
Vol 29 (03) ◽  
pp. 1950012
Author(s):  
Tianlong Ma ◽  
Jinling Wang ◽  
Mingzu Zhang
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Kronecker Product ◽  
Sufficient Condition ◽  
Connected Component ◽  
Edge Connectivity ◽  
Product Graphs ◽  
Minimum Number ◽  
Vertex Set ◽  
Product Of Graphs

The restricted edge-connectivity of a connected graph [Formula: see text], denoted by [Formula: see text], if exists, is the minimum number of edges whose deletion disconnects the graph such that each connected component has at least two vertices. The Kronecker product of graphs [Formula: see text] and [Formula: see text], denoted by [Formula: see text], is the graph with vertex set [Formula: see text], where two vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. In this paper, it is proved that [Formula: see text] for any graph [Formula: see text] and a complete graph [Formula: see text] with [Formula: see text] vertices, where [Formula: see text] is minimum edge-degree of [Formula: see text], and a sufficient condition such that [Formula: see text] is [Formula: see text]-optimal is acquired.

On Topological Invariants of the Product of Graphs

1969 ◽  
Vol 12 (2) ◽  
pp. 157-166 ◽  
Author(s):  
M. Behzad ◽  
S. E. Mahmoodian
Keyword(s):  
Cartesian Product ◽  
Topological Invariants ◽  
Point Sets ◽  
Undirected Graphs ◽  
Set Of Points ◽  
Product Of Graphs

We consider ordinary graphs, that is, finite, undirected graphs with no loops or multiple lines. The product (also called cartesian product [4]) G1 × G2 of two graphs G1 and G2 with point sets V1 and V2, respectively, has the cartesian product V1 × V2 as its set of points. Two points (u1, u2) and (v1, v2) are adjacent if u1 = v1 and u2 is adjacent with v2 or u2 = v2 and u1 is adjacent with v1.

scholarly journals The thickness of the Kronecker product of graphs

2020 ◽  
Vol 18 (2) ◽  
pp. 339-357
Author(s):  
Xia Guo ◽  
Yan Yang
Keyword(s):  
Kronecker Product ◽  
Product Of Graphs

SUPER EDGE CONNECTIVITY OF KRONECKER PRODUCTS OF GRAPHS

2014 ◽  
Vol 25 (01) ◽  
pp. 59-65 ◽  
Author(s):  
XIANG-LAN CAO ◽  
ELKIN VUMAR
Keyword(s):  
Connected Graph ◽  
Kronecker Product ◽  
Minimum Degree ◽  
Kronecker Products ◽  
Edge Connectivity ◽  
Vertex Set ◽  
Product Of Graphs

A graph G is said to be super edge connected (in short super – λ) if every minimum edge cut isolates a vertex of G. The Kronecker product of graphs G and H is the graph with vertex set V(G × H) = V(G) × V(H), where two vertices (u1, v1) and (u2, v2) are adjacent in G × H if u1u2 ∈ E(G) and v1v2 ∈ E(H). Let G be a connected graph, and let δ(G) and λ(G) be the minimum degree and the edge-connectivity of G, respectively. In this paper we prove that G × Kn is super-λ for n ≥ 3, if λ(G) = δ(G) and G ≇ K2. Furthermore, we show that K2 × Kn is super-λ when n ≥ 4. Similar results for G × Tn are also obtained, where Tn is the graph obtained from Kn by adding a loop to every vertex of Kn.

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