Introduction Software implementations for graph structures can represent a graph as an adjacency list or a sparse adjacency matrix. There are no standards regarding the implementation details of adjacency list or adjacency matrix structures. Notable parties propose different implementations for adjacency lists [1]. Guido van Rossum suggests using a hash table with arrays, where the arrays indic...

given a graph $g$, let $g^sigma$ be an oriented graph of $g$ with the orientation $sigma$ and skew-adjacency matrix $s(g^sigma)$. then the spectrum of $s(g^sigma)$ consisting of all the eigenvalues of $s(g^sigma)$ is called the skew-spectrum of $g^sigma$, denoted by $sp(g^sigma)$. the skew energy of the oriented graph $g^sigma$, denoted by $mathcal{e}_s(g^sigma)$, is defined as the sum of the n...

This paper describes a spectral method for graph-matching. We adopt a graphical models viewpoint in which the graph adjacency matrix is taken to represent the transition probability matrix of a Markov chain. The nodeorder of the steady state random walk associated with this Markov chain is determined by the co-efficent order of the leading eigenvector of the adjacency matrix. We match nodes in ...

Let G be a graph with adjacency matrix A and let F be a field. An F-matrix Q is a support matrix of G if A = [Q ̸=O], the zero-nonzero pattern of Q. If G has an invertible skew-symmetric support F-matrix S, the S-dual G of G is defined as the graph with adjacency matrix [S−1 ̸= O]. An analogous adjacency matrix dual, G has been examined in the literature for those bipartite graphs G with unique p...

These notes are not necessarily an accurate representation of what happened in class. The notes written before class say what I think I should say. I sometimes edit the notes after class to make them way what I wish I had said. There may be small mistakes, so I recommend that you check any mathematically precise statement before using it in your own work. These notes were last revised on Septem...

Characterization of singular graphs can be reduced to the non-trivial solutions of a system of linear homogeneous equations Ax = 0 for the 0-1 adjacency matrix A. A graph G is singular of nullity η(G) ≥ 1, if the dimension of the nullspace ker(A) of its adjacency matrix A is η(G). Necessary and sufficient conditions are determined for a graph to be singular in terms of admissible induced subgra...

When the graph is a path with n vertices, we simply call PG the path polynomial and denote it by Pn. Define An as the adjacency matrix of a path on n vertices. For several interesting classes of graphs, A(Gi) is a polynomial in A(G), where Gi is the ith distance graph of G ([5]). Actually, for distance-regular graphs, A(Gi) is a polynomial in A(G) of degree i, and this property characterizes th...

The energy of a molecular graph G is defined as the summation of the absolute values of the eigenvalues of adjacency matrix of a graph G. In this paper, an infinite class of fullerene graphs with 10n vertices, n ≥ 2, is considered. By proving centrosymmetricity of the adjacency matrix of these fullerene graphs, a lower bound for its energy is given. Our method is general and can be extended to ...

The characterization of a graph via the variable adjacency matrix enables to de ne a partially ordered relation on the walks. Studying the incidence algebra on this poset reveals unsuspected relations between connected and self-avoiding walks on the graph. These relations are derived by considering truncated versions of the characteristic polynomial of variable adjacency matrix, resulting in a ...