Critical ideals of signed graphs with twin vertices

This paper studies critical ideals of graphs with twin vertices, which are vertices with the same neighbors. A pair of such vertices are called replicated if they are adjacent, and duplicated, otherwise. Critical ideals of graphs having twin vertices have good properties and show regular patterns. Given a graph G=(V,E) and d is an element of Z(vertical bar V vertical bar), let G(d) be the graph obtained from G by duplicating dv times or replicating -d(v) times the vertex v when d(v)>0 or d(v)<0, respectively. Moreover, given delta is an element of{0,1,-1}(vertical bar V vertical bar), let T-delta(G)={G(d):d<1/4>Z(vertical bar V vertical bar) such that d(v)=0 if and only if delta(v)=0 and d(v)delta(v)>0 otherwise} be the set of graphs sharing the same pattern of duplication or replication of vertices. More than one half of the critical ideals of a graph in T-delta(G) can be determined by the critical ideals of G. The algebraic co-rank of a graph G is the maximum integer i such that the i-{\it th} critical ideal of G is trivial. We show that the algebraic co-rank of any graph in T-delta(G) is equal to the algebraic co-rank of G(delta). For a large enough d is an element of Z(V(G)), we show that the critical ideals of Gd have similar behavior to the critical ideals of the disjoint union of G and some set {K-nv}({v is an element of V(G)vertical bar dv<0}) of complete graphs and some set {T-nv}({v is an element of V(G)vertical bar dv>0}) of trivial graphs. Additionally, we pose important conjectures on the distribution of the algebraic co-rank of the graphs with twins vertices. These conjectures imply that twin-free graphs have a large algebraic co-rank, meanwhile a graph having small algebraic co-rank has at least one pair of twin vertices.