Given a bipartite graph G with bipartition (U,W), we denote by Q(G) the set of all real U×W matrices B=bu,w with bu,w=0 if u and w are non-adjacent, bu,w≠0 if u and w are connected by a single edge, ...and bu,w∈R if u and w are connected by multiple edges. We denote by N(G) the set of all U×W matrices X=xu,w with xu,w=0 if u and w are adjacent. We say that a matrix B∈Q(G) has the Asymmetric Strong Arnold Property (ASAP) if for all X∈N(G), if XTB=0 and BXT=0, then X=0.
If G is a bipartite graph for which there exists a matrix B∈Q(G) that has the Asymmetric Strong Arnold Property, we define the stable minimum bipartite rank mbrS(G) as the smallest rank of any matrix B∈Q(G) having the ASAP. We show that if H is a matching minor of G, then mbrS(G)≤mbrS(H)+1/2(|V(G)|−|V(H)|). If G has a bipartition with parts of the same size, we define the stable maximum bipartite nullity MbS(G) as the largest nullity of any matrix B∈Q(G) having the ASAP. Then MbS(H)≤MbS(G). We give a characterization in terms of forbidden matching minors of the classes of graphs G with MbS(G)=0 and with MbS(G)≤1.
Most modern legal systems have been searching for a legal measure that would help to imply omitted terms as well as rescue the validity of unsuccessfully concluded agreements, and they found such a ...solution in the concept of supplementary interpretation of contract, an example of which is conversio actus iuridici. The tendencies visible in genere prove that whenever the law provides for too rigorous requirements for private ordering, the idea to supplement, imply, or convert contracts occurs. The presented comparative legal perspective is of great importance for Polish jurisprudence, which generally rejects the concept of supplementary interpretation of contracts, but accepts conversio actus iuridici. The analysis presented in this paper encourages representatives of Polish jurisprudence to wider adoption of the idea of supplementary interpretation of contracts, following the example of foreign legal orders.
In this paper, we completely characterize all the trees on n vertices with diameter d for which there is a symmetric matrix with nullity n−d and n−d−1, respectively. These characterizations cover all ...recent results proved for the standard 0−1 adjacency matrices. Here a new technique is developed for the general case, breaking the limitation of star complement technique for the standard 0−1 adjacency matrices.
Let T be a tree with p≥3 pendant edges and let L(T) be its line graph. In this paper, we prove the multiplicity of an arbitrary adjacency eigenvalue of L(T) is no larger than p−1. Furthermore, the ...line graphs L(T) of trees which contain an adjacency eigenvalue with multiplicity p−1 are completely determined.
Signed graphs with maximum nullity two Arav, Marina; Dahlgren, F. Scott; van der Holst, Hein
Linear algebra and its applications,
10/2023, Letnik:
675
Journal Article
Recenzirano
A signed graph is a pair (G,Σ), where G=(V,E) is a graph (in which parallel edges are permitted, but loops are not) with V={1,…,n} and Σ⊆E. The edges in Σ are called odd and the other edges of E ...even. If there are parallel edges, then only two edges in each parallel class are permitted, one of which is even and one of which is odd. By S(G,Σ) we denote the set of all symmetric n×n matrices A=ai,j with ai,j<0 if i and j are connected by an even edge, ai,j>0 if i and j are connected by an odd edge, ai,j∈R if i and j are connected by both an even and an odd edge, ai,j=0 if i≠j and i and j are non-adjacent, and ai,i∈R for all vertices i.
The maximum nullity M(G,Σ) of a signed graph (G,Σ) is the maximum nullity attained by any A∈S(G,Σ). Arav et al. gave a combinatorial characterization of 2-connected signed graphs (G,Σ) with M(G,Σ)=2. In this paper, we give a complete combinatorial characterization of the signed graphs (G,Σ) with M(G,Σ)=2.
In this paper the problem of the existence of regular nut graphs is addressed. A generalization of Fowler’s Construction which is a local enlargement applied to a vertex in a graph is introduced to ...generate nut graphs of higher order. Let N(ρ) denote the set of integers n such that there exists a regular nut graph of degree ρ and order n. It is proven that N(3) = {12} ∪ {2k : k ≥ 9} and that N(4) = {8, 10, 12} ∪ {n : n ≥ 14}. The problem of determining N(ρ) for ρ > 4 remains completely open.
Prezentul demers științific este consacrat unor probleme ce țin de clarificarea conceptului de natură juridică a conversiunii. Se menționează că această ipotetică instituție juridică nu este ...analizată în mod uniform de către doctrina statelor lumii, dar nici nu are o prezentare precisă în legislațiile diferitor state. Mai mult, există sisteme de drept care nu o consacră expres ori tacit. În această ordine de idei, accentul s-a plasat pe necesitatea de „vindecare” juridică a actelor lovite de nulitate absolută, în special aducându-se ca exemplu soluțiile optime identificate de către legiuitori și doctrinari privind aplicarea acestei construcții juridice. Se demonstrează că este importantă, deși extrem de dificilă, implementarea acestui mecanism juridic la stadiul actual al formatului normativ național, atât timp cât natura juridică a conversiunii rămâne incertă. Drept un imperativ al cercetării științifice se impune necesitatea de coagulare a tuturor viziunilor privind natura juridică a conversiunii, în vederea obținerii unei optici uniforme. Se demonstrează multipolaritatea naturii juridice a conversiunii, privită uneori ca modalitate de îndreptare a cauzelor de nulitate, alterori ca excepție de la principiul quod nullum est, nullum producit effectum ori ca modalitate de înlăturare a efectelor nulității actului juridic civil, fie ca principiu care înlătura regula quod nullum est, nullum producit effectum ori ca principiu care menține efectele actului lovit de nulitate. Acestea rămân a fi doar câteva dintre soluțiile date de doctrină, care nu s-a focusat cu precădere pe acest segment fragil. Se menționează că legea civilă a Republicii Moldova nu acoperă reglementarea juridică a conversiunii, deși anumite deducții pot fi efectuate. Se demonstrează că conversiunea are o importanță remarcabilă într-un sistem de drept și că introducerea unor reglementări legale exprese nu reprezintă doar un exercițiu intelectiv, dar și o necesitate juridică. THE CIVIL LEGAL DIMENSION OF THE CONVERSION. COMPARATIVE LAWThis scientific research is devoted to issues related to the clarification of the legal concept of conversion. It has been mentioned that this hypothetical legal institution is not uniformly analyzed by the doctrine of the states of the world, nor does it have a precise presentation in the laws of different states. Moreover, there are systems of law that do not expressly or tacitly consecrate it. In this context, emphasis was placed on the need for legal "healing" of acts of absolute nullity, in particular by exemplifying the optimal solutions identified by legislators and doctrines regarding the application of this legal construction. It has been demonstrated that it is important, although extremely difficult, to implement this legal mechanism at the current state of the national regulatory format, as long as the legal nature of the conversion remains uncertain. An imperative of scientific research has placed the need to coagulate all the views on the legal nature of conversion in order to obtain a uniform optician. It has been shown the multipolarity of the legal nature of conversion, which is sometimes seen as a way of redressing the causes of nullity, as well as exceptions to the principle of nullum est, nullum produit effectum, or as a way of removing the effects of the nullity of the civil legal act, or a principle that removes the rule quod nullum est, nullum produit effectum, or the principle that maintains the effects of the act of nullity, and these remain only a few of the solutions given by the doctrine, which was not particularly focused on this fragile segment. It was mentioned that the civil law of the Republic of Moldova does not cover the juridical regulation of the conversion, although certain deductions can be made, as it was stated in the present scientific research. It has been shown that conversion has a remarkable importance in a legal system and that the introduction of explicit legal regulations is not only an intellectual exercise but also a legal necessity.
By |V(G)|, |E(G)|, η(G), and m(G) we denote respectively the order, the number of edges, the nullity, and the matching number of a (simple) graph G. Recently Wang and Wong have proved that for every ...graph G, η(G)≤|V(G)|−2m(G)+2c(G), where c(G)=|E(G)|−|V(G)|+θ(G), θ(G) being the number of connected components of G. In this paper graphs G that satisfy the equality η(G)=|V(G)|−2m(G)+2c(G) are characterized.
In this paper, we completely characterize those trees on n vertices for which there is a singular matrix with nullity k and the number of P-vertices is n−k−1 or n−k−2. The characterization of acyclic ...matrices, with rank r and the number of P-vertices is r−1, or with odd rank r and the number of P-vertices is r−2, was first investigated in Fonseca et al. (2021) 10. Here we introduce a unified method to revisit those results, and further cover the unknown case with even rank r and the number of P-vertices being r−2. In the end, a continuity problem is fully solved.
A graph G is said to be determined by its permanental spectrum if any graph having the same permanental spectrum as G is isomorphic to G. In this paper, we introduce the permanental nullity of a ...graph, the multiplicity of the number zero in the permanental spectrum of a graph, to study graphs determined by their permanental spectra. First, we determine all graphs of order n whose permanental nullities are n−2, n−3, n−4 and n−5, respectively. Then, we show that all graphs with the permanental nullity n−2, n−3, or n−5, and all non-bipartite graphs with the permanental nullity n−4 are determined by their permanental spectra. In particular, we prove that the complete bipartite graphs are determined by their permanental spectra.
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