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1.
Wiener index of strong product of graphs
Iztok Peterin, Petra Žigert Pleteršek, 2018, original scientific article

Abstract: The Wiener index of a connected graph ▫$G$▫ is the sum of distances between all pairs of vertices of ▫$G$▫. The strong product is one of the four most investigated graph products. In this paper the general formula for the Wiener index of the strong product of connected graphs is given. The formula can be simplified if both factors are graphs with the constant eccentricity. Consequently, closed formulas for the Wiener index of the strong product of a connected graph ▫$G$▫ with a cycle are derived.
Keywords: Wiener index, graph product, strong product
Published: 30.11.2017; Views: 750; Downloads: 302
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2.
On the rainbow connection of Cartesian products and their subgraphs
Sandi Klavžar, Gašper Mekiš, 2012, original scientific article

Abstract: Rainbow connection number of Cartesian products and their subgraphs are considered. Previously known bounds are compared and non-existence of such bounds for subgraphs of products are discussed. It is shown that the rainbow connection number of an isometric subgraph of a hypercube is bounded above with the rainbow connection number of the hypercube. Isometric subgraphs of hypercubes with the rainbow connection number smaller as much as possible than the rainbow connection of the hypercube are constructed. The concept of c-strong rainbow coloring is introduced. In particular it is proved that the so-called ▫$\Theta$▫-coloring of an isometric subgraph of a hypercube is its unique optimal c-strong rainbow coloring.
Keywords: graph theory, rainbow connection, strong rainbow connection, Cartesian product of graphs, isometric subgraph, hypercube
Published: 31.03.2017; Views: 607; Downloads: 286
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3.
Edge-connectivity of strong products of graphs
Boštjan Brešar, Simon Špacapan, 2007, original scientific article

Abstract: The strong product ▫$G_1 \boxtimes G_2$▫ of graphs ▫$G_1$▫ and ▫$G_2$▫ is the graph with ▫$V(G_1) \times V(G_2)$▫ as the vertex set, and two distinct vertices ▫$(x_1,x_2)$▫ and ▫$(y_1,y_2)$▫ are adjacent whenever for each ▫$i\in \{1,2\}$▫ either ▫$x_i=y_i$▫ or ▫$x_iy_i \in E(G_i)$▫. In this note we show that for two connected graphs ▫$G_1$▫ and ▫$G_2$▫ the edge-connectivity ▫$\lambda(G_1 \boxtimes G_2)$▫ equals ▫$\min\{\delta(G_1\boxtimes G_2), \lambda(G_1)(|V(G_2)|+2|E(G_2)|), \lambda(G_2)(|V(G_1)|+2|E(G_1)|)\}$▫. In addition, we fully describe the structure of possible minimum edge cut sets in strong products of graphs.
Keywords: mathematics, graph theory, connectivity, strong product, graph product, separating set
Published: 31.03.2017; Views: 659; Downloads: 266
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4.
Intervals and convex sets in strong product of graphs
Iztok Peterin, 2013, original scientific article

Abstract: Obravnavamo intervale in konveksne množice krepkega produkta. Vozlišča poljubnega intervala iz ▫$G boxtimes H$▫ so klasificirana z najkrajšimi potmi v enem faktorju in s sprehodi v dugem rahlo modificiranem faktorju. Konveksne množice krepkega produkta so karakterizirane s konveksnostjo obeh projekcij in še tremi lokalnimi lastnostmi, med katerimi je tudi 2-konveksnost.
Keywords: teorija grafov, krepki produkt, geodetska konveksnost, interval, graph theory, strong product, geodesic convexity, interval
Published: 10.07.2015; Views: 490; Downloads: 84
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5.
On the b-chromatic number of some graph products
Marko Jakovac, Iztok Peterin, 2012, original scientific article

Abstract: Pravilno barvanje vozlišč grafa kjer vsak barvni razred vsebuje vozlišče, ki ima soseda v vseh preostalih barvnih razredih, imenujemo b-barvanje. Največje naravno število ▫$varphi (G)$▫, za katero obstaja b-barvanje grafa ▫$G$▫, imenujemo b-kromatično število. Določimo nekatere spodnje in zgornje meje b-kromatičnega števila za krepki produkt ▫$G,boxtimes, H$▫, leksikografski produkt ▫$G[H]$▫ in za direktni produkt ▫$G,times, H$▫. Prav tako določimo nekatere točne vrednosti za produkte poti, ciklov, zvezd in polnih dvodelnih grafov. Pokažemo tudi, da lahko določimo b-kromatično število za ▫$P_n ,boxtimes, H$▫, ▫$C_n ,boxtimes, H$▫, ▫$P_n[H]$▫, ▫$C_n[H]$▫ in ▫$K_{m,n}[H]$▫ za poljuben graf ▫$H$▫, če sta le ▫$m$▫ in ▫$n$▫ dovolj veliki.
Keywords: teorija grafov, b-kromatično število, krepki produkt, leksikografski produkt, direktni produkt, graph theory, b-chromatic number, strong product, lexicographic product, direct product
Published: 10.07.2015; Views: 559; Downloads: 71
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6.
NZ-flows in strong products of graphs
Wilfried Imrich, Iztok Peterin, Simon Špacapan, Cun-Quan Zhang, 2010, original scientific article

Abstract: Za krepki produkt ▫$G_1 boxtimes G_2$▫ grafov ▫$G_1$▫ in ▫$G_2$▫ dokažemo, da je ▫${mathbb{Z}}_3$▫-pretočno kontraktibilen natanko tedaj, ko ▫$G_1 boxtimes G_2$▫ ni izomorfen ▫$Tboxtimes K_2$▫ (kar poimenujemo ▫$K_4$▫-drevo), kjer je ▫$T$▫ drevo. Sledi, da za ▫$G_1 boxtimes G_2$▫ obstaja NZ 3-pretok, razen če je ▫$G_1 boxtimes G_2$▫ ▫$K_4$▫-drevo. Dokaz je konstruktiven in implicira polinomski algoritem, ki nam vrne NZ 3-pretok, če ▫$G_1 boxtimes G_2$▫ ni ▫$K_4$▫-drevo, oziroma NZ 4-pretok sicer.
Keywords: matematika, teorija grafov, celoštevilski pretoki, krepki produkt, poti, cikli, nikjer ničelni pretok, mathematics, graph theory, integer flows, strong product, paths, cycles
Published: 10.07.2015; Views: 502; Downloads: 100
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7.
Strong isometric dimension, biclique coverings, and Sperner's theorem
Dalibor Fronček, Janja Jerebic, Sandi Klavžar, Petr Kovář, 2007, original scientific article

Abstract: Krepka izometrična dimenzija grafa ▫$G$▫ je najmanjše število ▫$k$▫, tako da lahko ▫$G$▫ izometrično vložimo v krepki produkt ▫$k$▫ poti. Z uporabo Spernerjevega izreka je določena krepka izometrična dimenzija Hammingovih grafov ▫$K_2,{square}, K_n$▫.
Keywords: matematika, teorija grafov, krepka izometrična dimenzija, Hammingovi grafi, mathematics, graf theory, strong product, Hamming graphs
Published: 10.07.2015; Views: 545; Downloads: 57
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8.
Weak reconstruction of strong product graphs
Blaž Zmazek, Janez Žerovnik, 2007, original scientific article

Abstract: We prove that any nontrivial connected strong product graph can be uniquely reconstructed from each of its one vertex deleted subgraphs.
Keywords: matematika, teorija grafov, problem rekonstrukcije, šibka rekonstrukcija, krepki produkt grafov, mathematics, graph theory, reconstruction problem, strong product, composite graphs
Published: 10.07.2015; Views: 504; Downloads: 344
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9.
On induced and isometric embeddings of graphs into the strong product of paths
Janja Jerebic, Sandi Klavžar, 2006, original scientific article

Abstract: Primerjani sta krepka izometrična dimenzija in sosedna izometrična dimenzija grafov. Koncepta sta ekvivalentna za grafe premera 2 in v tem primeru se problem določitve dimenzije reducira na problem pokritja s polnimi dvodelnimi grafi. S pomočjo tega pristopa je določena krepka izometrična dimenzija in sosedna izometrična dimenzija za različne grafe (na primer za Petersenov graf). Podan je pozitiven odgovor na Problem 4.1 iz [Fitzpatrick, Nowakowski, The strong isometric dimension of finite reflexive graphs, Discuss. Math. Graph Theory 20 (2000) 23-38], ali obstaja tak graf ▫$G$▫, ki ima krepko izometrično dimenzijo večjo od ▫$lceil |V(G)|/2 rceil$▫.
Keywords: matematika, teorija grafov, krepki produkt grafov, krepka izometrična dimenzija, sosedna izometrična dimenzija, mathematics, graph theory, strong product of graphs, adjacent isometric dimension, strong isometric dimension
Published: 10.07.2015; Views: 653; Downloads: 69
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10.
On the weak reconstruction of strong product graphs
Blaž Zmazek, Janez Žerovnik, 2003, published scientific conference contribution

Abstract: We prove that any nontrivial connected strong product graph can be uniquely reconstructed from each of its one vertex deleted subgraphs.
Keywords: matematika, teorija grafov, problem rekonstrukcije, šibka rekonstrukcija, krepki produkt grafov, mathematics, graph theory, reconstruction problem, strong product, composite graphs
Published: 10.07.2015; Views: 527; Downloads: 61
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