On conjugate adjacency matrices of a graph

Majorisations for the eigenvectors of graph-adjacency matrices

We develop majorisation results that characterise changes in eigenvector components of a graph’s adjacency matrix when its topology is changed. Specifically, for general (weighted, directed) graphs, we characterise changes in dominant eigenvector components for singleand multi-row incrementations. We also show that topology changes can be tailored to set ratios between the components of the dom...

Adjacency Matrices That Are Incidence Matrices: 10967

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Spectra of signed adjacency matrices

A signed adjacency matrix is a {−1, 0, 1}-matrix A obtained from the adjacency matrix A of a simple graph G by symmetrically replacing some of the 1’s of A by −1’s. Bilu and Linial have conjectured that if G is k-regular, then some A has spectral radius ρ(A) ≤ 2 √ k − 1. If their conjecture were true then, for each fixed k > 2, it would immediately guarantee the existence of infinite families o...

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