Additionally, we make an assumption that Ai are non-empty sets, for i=0, 1, . . . ,m, where m=[n/2]. For each set of functions An we determine the number of meaningful compositions of higher order in implicit and explicit form. Let us define a binary relation ρ ”to be in composition” with ∇iρ∇j = 1 iff the composition ∇j ◦∇i is meaningful for i, j ∈ {1, 2, . . . , n}. Let us form an adjacency m...

A graph is called integral if all eigenvalues of its adjacency matrix are integers. In this paper, we investigate integral complete r-partite graphsKp1,p2,...,pr =Ka1·p1,a2·p2,...,as ·ps with s=3, 4.We can construct infinite many new classes of such integral graphs by solving some certain Diophantine equations. These results are different from those in the existing literature. For s = 4, we giv...

In this paper, only simple graphs are considered. A graphG is nonsingular if its adjacency matrix A(G) is nonsingular. A nonsingular graph G satisfies reciprocal eigenvalue property (property R) if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and G satisfies strong reciprocal eigenvalue property (property SR) if the reciprocal of each eigenvalue o...

The r-parity tensor of a graph is a generalization of the adjacency matrix, where the tensor’s entries denote the parity of the number of edges in subgraphs induced by r distinct vertices. For r = 2, it is the adjacency matrix with 1’s for edges and −1’s for nonedges. It is well-known that the 2-norm of the adjacency matrix of a random graph is O( √ n). Here we show that the 2-norm of the r-par...

This is an expository survey of the uses of matrices in the theory of simple graphs with signed edges. A signed simple graph is a graph, without loops or parallel edges, in which every edge has been declared positive or negative. For many purposes the most significant thing about a signed graph is not the actual edge signs, but the sign of each circle (or ‘cycle’ or ’circuit’), which is the pro...

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Problem 1. [20 points] The adjacency matrix A of a graph G with n vertices as defined in lecture is an n×n matrix in which Ai,j is 1 if there is an edge from i to j and 0 if there is not. In lecture we saw how the smallest k where Ai,j 6= 0 describes the length of the shortest path from i to j. Given a combinatorial interpretation of the following statements about the adjacency matrix in terms ...

Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example we show that the query complexity of Minimum Spanning Tree is in Θ(...