Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2006.00002

No signed graph with the nullity $η(G,σ)=|V(G)|-2m(G)+2c(G)-1$

Yong Lu·Jingwen Wu
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $G^σ=(G,σ)$ be a signed graph and $A(G,σ)$ be its adjacency matrix. Denote by $m(G)$ the matching number of $G$. Let $η(G,σ)$ be the nullity of $(G,σ)$. He et al. [Bounds for the matching number and cyclomatic number of a signed graph in terms of rank, Linear Algebra Appl. 572 (2019), 273--291] proved that $$|V(G)|-2m(G)-c(G)\leqη(G,σ)\leq |V(G)|-2m(G)+2c(G),$$ where $c(G)$ is the dimension of cycle space of $G$. Signed graphs reaching the lower bound or the upper bound are respectively characterized by the same paper. In this paper, we will prove that no signed graphs with nullity $|V(G)|-2m(G)+2c(G)-1$. We also prove that there are infinite signed graphs with nullity $|V(G)|-2m(G)+2c(G)-s,~(0\leq s\leq3c(G), s\neq1)$ for a given $c(G)$.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yong Lu, Jingwen Wu. 2020-06-15. No signed graph with the nullity $η(G,σ)=|V(G)|-2m(G)+2c(G)-1$. https://arxiv.org/abs/2006.00002

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗