Quantum Probability Aspects to Lexicographic and Strong Products of Graphs

Abstract

The adjacency matrix of the lexicographic product of graphs is decomposed into a sum of monotone independent random variables in a certain product state. The adjacency matrix of the strong product of graphs admits an expression in terms of commutative independent random variables in a product state. Their spectral distributions are obtained by using the monotone, classical and Mellin convolutions of probability distributions.KEYWORDS: adjacency matrix, convolution of probability distributions, lexicographic product, spectral distribution, strong product Products of GraphsA graph G ¼ ðV; EÞ is a pair, where V is a non-empty set of vertices and E a set of edges, i.e., a subset of unordered pairs of distinct vertices. If fx; yg 2 E, we say that x and y are adjacent and write x $ y. We deal with both finite and infinite graphs, but always assume that a graph is locally finite, i.e., degðxÞ < 1 for all vertices x 2 V. The adjacency matrix of G, denoted by A ¼ A½G, is a matrix with index set V Â V defined byGiven two graphs G 1 ¼ ðV 1 ; E 1 Þ and G 2 ¼ ðV 2 ; E 2 Þ there is a large variety of forming their product to obtain a larger graph, see e.g., [4] and references cited therein. From the quantum probability viewpoint we have so far studied the Cartesian, star, comb and free products of graphs [1,6,9,10]. In this paper, being based on a similar spirit, we will discuss the lexicographic and strong products of graphs, and derive their spectral distributions using certain concepts of independence in quantum probability. Definition 1.1. The lexicographic product of G 1 and G 2 , denoted by G 1 B L G 2 , is the graph on V ¼ V 1 Â V 2 , where two distinct vertices ðx 1 ; y 1 Þ and ðx 2 ; y 2 Þ are adjacent whenever (i) x 1 $ x 2 ; or (ii) x 1 ¼ x 2 and y 1 $ y 2 . Lemma 1.2. Let G 1 and G 2 be graphs with adjacency matrices A 1 and A 2 , respectively. Then the adjacency matrix of the lexicographic productwhere J 2 is the matrix with index set V 2 Â V 2 whose entries are all one, and I 1 is the identity matrix with index setThe proof is straightforward by definition and is omitted. The Cartesian product of G 1 and G 2 , denoted by G 1 Â C G 2 , is the graph on V ¼ V 1 Â V 2 , where two distinct vertices ðx 1 ; y 1 Þ and ðx 2 ; y 2 Þ are adjacent whenever (i) x 1 ¼ x 2 and y 1 $ y 2 ; or (ii) x 1 $ x 2 and y 1 ¼ y 2 . The adjacency matrix of G 1 Â C G 2 is given byð1:2ÞThe Cartesian product is associative and commutative. By definition, G 1 Â C G 2 is a subgraph of G 1 B L G 2 , which is viewed also from the adjacency matrices ð1:1Þ and ð1:2Þ.Definition 1.3. The strong product of G 1 and G 2 , denoted by G 1 Â S G 2 , is the graph on V ¼ V 1 Â V 2 , where two distinct vertices ðx 1 ; y 1 Þ and ðx 2 ; y 2 Þ are adjacent whenever (i) x 1 ¼ x 2 or x 1 $ x 2 ; and (ii) y 1 ¼ y 2 or y 1 $ y 2 .