Several researchers have recently explored various graph parameters that can or cannot be characterized by the spectrum of a matrix associated with a graph. In this paper, we show that several ...NP-hard zero forcing numbers are not characterized by the spectra of several types of associated matrices with a graph. In particular, we consider standard zero forcing, positive semidefinite zero forcing, and skew zero forcing and provide constructions of infinite families of pairs of cospectral graphs, which have different values for these numbers. We explore several methods for obtaining these cospectral graphs including using graph products, graph joins, and graph switching. Among these, we provide a construction involving regular adjacency cospectral graphs; the regularity of this construction also implies cospectrality with respect to several other matrices including the Laplacian, signless Laplacian, and normalized Laplacian. We also provide a construction where pairs of cospectral graphs can have an arbitrarily large difference between their zero forcing numbers.
This essay examines the controversial usage of a procedure known as the épreuve du congrès to adjudicate suits by women seeking to annul their marriages in French Church courts (officialités) on the ...grounds of their husbands’ impotence. Drawing on an analysis of thirty-two cases reconstituted from the Officialité de Paris in the early seventeenth century, this article assesses the extent to which jurists’ and medical experts’ defenses and critiques of the congrès as a method of proof corresponded to the ways it was used in practice.
Abstract Let $f\colon M^{2n}\to \mathbb {R}^{2n+p}$ , $2\leq p\leq n-1$ , be an isometric immersion of a Kaehler manifold into Euclidean space. Yan and Zheng (2013, Michigan Mathematical Journal 62, ...421–441) conjectured that if the codimension is $p\leq 11$ , then, along any connected component of an open dense subset of $M^{2n}$ , the submanifold is as follows: it is either foliated by holomorphic submanifolds of dimension at least $2n-2p$ with tangent spaces in the kernel of the second fundamental form whose images are open subsets of affine vector subspaces, or it is embedded holomorphically in a Kaehler submanifold of $\mathbb {R}^{2n+p}$ of larger dimension than $2n$ . This bold conjecture was proved by Dajczer and Gromoll just for codimension 3 and then by Yan and Zheng for codimension 4. In this paper, we prove that the second fundamental form of the submanifold behaves pointwise as expected in case that the conjecture is true. This result is a first fundamental step for a possible classification of the nonholomorphic Kaehler submanifolds lying with low codimension in Euclidean space. A counterexample shows that our proof does not work for higher codimension, indicating that proposing $p=11$ in the conjecture as the largest codimension is appropriate.
Let G be a simple graph with n(G) vertices and e(G) edges. Denoted by η(G) and m⁎(G) the nullity and the fractional matching number of G, respectively. The dimension of cycle space of G is defined as ...c(G)=e(G)−n(G)+ω(G), where ω(G) denotes the number of connected components of G.
In this paper, we prove that n(G)−2m⁎(G)≤η(G)≤n(G)−2m⁎(G)+2c(G) for a graph G, which improves the main results of Wang and Wong (2014) 19 and Ma and Fang (2019) 11, respectively. Furthermore, all graphs with nullity η(G)=n−2m⁎(G)+2c(G) are determined. We also prove that there is no graph with nullity η(G)=n−2m⁎(G)+2c(G)−1; and for fixed c(G), infinitely many connected graphs with nullity n−2m⁎(G)+2c(G)−k(0≤k≤2c(G),k≠1) are also constructed. As an application of the above results, we also prove that if G is nonsingular, then G has a fractional perfect matching.
The nullity η(G) of G is the multiplicity of 0 as an eigenvalue of A(G). In this paper, we completely solve the following conjecture proposed by Zhou, Wong and Sun in Linear Algebra and its ...Applications, 555 (2018) 314-320: Let G be a connected graph of order n with nullity η(G) and the maximum degree Δ≥2. Thenη(G)≤(Δ−2)n+2Δ−1, the equality holds if and only if G≅Cn (n≡0(mod4)) or G≅KΔ,Δ.
This research paper is based on the investigation of over 1000 lawsuits for the declaration of nullity of mar- riage initiated at the Church Interdiocesan Tribunal in Split between 2000 and 2020. The ...main aim of this research paper was to focus on the lawsuits connected with the canon 1095, n. 3, especially on the problems of gambling addiction. The materials were analysed synthetically, and data were collected on a number of lawsuits conducted under this canon and on specific reasons for filing a lawsuit. Data on annulation of matrimony were also collected. The data are presented in an anonymous and aggregated manner. Descriptive and inferential statistical methods were used. The paper consists of a theoretical part and methodology, which includes the motive of the paper, the description of the sample used and the criteria of inclusion, materials, methods and procedures of paper, and the presentation of the results, together with the discussion of the results.
Let G be a simple graph of order n. The nullity of a graph G, denoted by η(G), is the multiplicity of 0 as an eigenvalue of its adjacency matrix. If G has at least one cycle, then the girth of G, ...denoted by gr(G), is the length of the shortest cycle in G. It is known that η(G) is bounded above by n−gr(G)+2 if 4|gr(G) and by n−gr(G) if 4∤gr(G). In this paper it is proved that when G is connected, η(G)=n−gr(G)+2 if and only if G is a complete bipartite graph, different from a star, or a cycle of length a multiple of 4; that if G is not a complete bipartite graph or a cycle of length a multiple of 4, then η(G)≤n−gr(G). Connected graphs of order n with girth g and nullity n−g are characterized. This work also settles the problem of characterizing connected graphs with rank equal to girth and the problem of identifying all connected graphs G that contains a given nonsingular cycle as a shortest cycle and with the same rank as G.
It is known that the zero forcing number of a graph is an upper bound for the maximum nullity of the graph (see AIM Minimum Rank - Special Graphs Work Group (F. Barioli, W. Barrett, S. Butler, S. ...Cioab$\breve{\text{a}}$, D. Cvetkovi$\acute{\text{c}}$, S. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanovi$\acute{\text{c}}$, H. van der Holst, K. Vander Meulen, and A. Wangsness). Linear Algebra Appl., 428(7):1628--1648, 2008). In this paper, we search for characteristics of a graph that guarantee the maximum nullity of the graph and the zero forcing number of the graph are the same by studying a variety of graph parameters that give lower bounds on the maximum nullity of a graph. Inparticular, we introduce a new graph parameter which acts as a lower bound for the maximum nullity of the graph. As a result, we show that the Aztec Diamond graph's maximum nullity and zero forcing number are the same. Other graph parameters that are considered are a Colin de Verdiére type parameter and vertex connectivity. We also use matrices, such as a divisor matrix of a graph and an equitable partition of the adjacency matrix of a graph, to establish a lower bound for the nullity of the graph's adjacency matrix.
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