n-3. Also, we introduce some graphs with rank of 4 such that these two parameters are equal.Keywords: zero forcing number,聽maximum nullity,聽minimum rank,聽rankAdditional informationAuthor informationEbrahim VatandoostEbrahim Vatandoost is presently working as an assistant professor in the ...
The rank of a graph is defined to be the rank of its adjacency matrix and the nullity of a graph is defined to be the multiplicity of zero eigenvalues of its adjacency matrix. They are respectively denoted by r(G) and η(G). These two concepts are equivalent in the sense that r(G)...
A natural problem is: How about the relation between the skew-rank of an oriented graphGσand the rank of its underlying graph? In this paper, we focus our attention on this problem. Denote by d(G) the dimension of cycle spaces of G, that is d(G)=|E(G)|−|V(G)|+θ(G), ...
Homework Equations Let ##\sigma:U\rightarrow V## and ##\tau:V\rightarrow W## such that ##dimU=n##, ##dimV=m##. Define ##v(\tau)## to be the nullity of ##\tau##... Terrell Thread Feb 22, 2018 Linear algebraLinear transformationsrankTransformations ...
{\mathcal {l}}\cap m(a^n), \end{aligned}$$ and the rank-nullity theorem yields $$\begin{aligned} \dim {\text {span}}\{[x_{1,n}],\ldots ,[x_{m,n}]\}&= \dim r(t)= m- \dim n(t) \\&= m-(\dim n(s) + \dim {\mathcal {l}}\cap m(a^n)) \\&= \dim ...
Triangle-free graph Rank Congruent vertex Nullity Mathematics Subject Classification 05C50 Access this article Log in via an institution Subscribe and save Springer+ Basic €32.70 /Month Get 10 units per month Download Article/Chapter or eBook 1 Unit = 1 Article or 1 Chapter Cancel anytime Sub...
In this section we give a lower and an upper bounds for the skew-rank of a connected oriented graph in terms of its matching number and its dimension of the cycle spaces. Theorem 3.1 (i) Let Gσ be a connected oriented graph. Then 2m(G)−2β(G)≤sr(Gσ)≤2m(G), where m(G...
In this paper, we concentrate on the relation between the H-rank of G˜ and the rank of G. We first show that −2d(G)⩽rk(G˜)−r(G)⩽2d(G) for every mixed graph G˜. Then we characterize all the mixed graphs that attain the above lower (resp. upper) bound. By ...
Let Gσ be an oriented graph with skew adjacency matrix S(Gσ). The skew energy Es(Gσ) of Gσ is the sum of the norms of all eigenvalues of S(Gσ) and the skew rank rs(Gσ) of Gσ is the rank of S(Gσ). Recently, Tian and Wong gave a lower bound of Es(Gσ) in te...
Parameters related to tree-width, zero forcing, and maximum nullity of a graph 2013, Journal of Graph Theory View all citing articles on Scopus ☆ The workshop “Spectra of Families of Matrices described by Graphs, Digraphs, and Sign Patterns”, was held at the American Institute of Mathematic...