On graphs with signed inverses

Abstract

A graph G is called invertible if its adjacency matrix A has an inverse which is the adjacency matrix of some graph H. All such graphs were shown by Haraq and Minc to have the form nKz. We now introduce signed invertible (or briefly s-invertible) graphs G as those whose inverse H is a signed graph. We identify two infinite classes of s-invertible graphs: the paths PZn of even order, and the corona of any graph with KI. We then characterize s-invertible trees. INVERTIBLE GRAPHSLet A = A(G) be the adjacency matrix of a graph G . The matrix A may or may not have an inverse. When it does, we say that graph G is invertible if A-' is the adjacency matrix of some graph H . In this case, H is called the inverse graph of G . Obviously, whether G is invertible is independent of the particular labeling of the vertices used to obtain A.If A(G) has an inverse which is the adjacency matrix of a signed graph H , we say that G is signed invertible, or briefly s-invertible, and H is the signed inverse of C .Clearly, H is also independent of the labeling used for G. Our purpose is to study invertible and s-invertible graphs. We follow the terminology and notation of the book Obviously, an integral matrix M has an integral inverse if and only if det M = 2 1.Therefore,the graphs we are considering must satisfy det A = k 1. If two nonadjacent vertices in a graph have the same neighborhood, then det A = 0 as two rows of A would be identical. Thus, a complete bipartite graph Km," has zero [31.

Inverses of triangular matrices and bipartite graphs

“…Our proof technique is different. In the same spirit, Theorem 1 leads to a formula for the inverse of the adjacency matrix of a weighted tree (see Section 4) when the tree has a perfect matching, generalizing a well-known result from [4,7] (see also [1,Section 3.6]).…”

On nonsingular trees and a reciprocal eigenvalue property

“…Since T is not a corona tree, it has a matching edge, say ½3, 4, whose endvertices are not pendants. Thus it has a path of length 5, say, [1,2,3,4,5,6] such that the edges [1,2], [3,4], [5,6] 2 M: Note that T 0 ¼ T ½3, 4 is either a corona tree or not.…”

Unicyclic graphs with bicyclic inverses

“…The pseudo-inverse graph P I(G) of G is the graph, defined on the same vertex set as G, in which the vertices x and y are adjacent if and only if G − x − y has a perfect matching. In 1988, Buckley, Doty and Harary introduced in [3] the signed inverse of a graph. A signed graph is a graph in which each edge has a positive or negative sign, see [7].…”