The Adjacency Matroid of a Graph

The Adjacency Matroid of a Graph

If G is a looped graph, then its adjacency matrix represents a binary matroid MA(G) on V (G). MA(G) may be obtained from the delta-matroid represented by the adjacency matrix of G, but MA(G) is less sensitive to the structure of G. Jaeger proved that every binary matroid is MA(G) for some G [Ann. Discrete Math. 17 (1983), 371-376]. The relationship between the matroidal structure of MA(G) and t...

Adjacency metric dimension of the 2-absorbing ideals graph

Let Γ=(V,E) be a graph and ‎W_(‎a)={w_1,…,w_k } be a subset of the vertices of Γ and v be a vertex of it. The k-vector r_2 (v∣ W_a)=(a_Γ (v,w_1),‎…‎ ,a_Γ (v,w_k)) is the adjacency representation of v with respect to W in which a_Γ (v,w_i )=min{2,d_Γ (v,w_i )} and d_Γ (v,w_i ) is the distance between v and w_i in Γ. W_a is called as an adjacency resolving set for Γ if distinct vertices of ...

Regions Adjacency Graph Applied

The homology of the cycle matroid of a coned graph

The cone Ĝ of a finite graph G is obtained by adding a new vertex p, called the cone point, and joining each vertex of G to p by a simple edge. We show that the rank of the reduced homology of the independent set complex of the cycle matroid of Ĝ is the cardinality of the set of the edge-rooted forests in the base graph G. We also show that there is a basis for this homology group such that the...

The Adjacency Matrix and Graph Coloring

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...