Maxima of the Q-index: forbidden 4-cycle and 5-cycle

This paper gives tight upper bounds on the largest eigenvalue q(G) of the signless Laplacian of graphs with no 4-cycle and no 5-cycle. If n is odd, let F_{n} be the friendship graph of order n; if n is even, let F_{n} be F_{n-1} with an edge hanged to its center. It is shown that if G is a graph of order n, with no 4-cycle, then q(G)<q(F_{n}), unless G=F_{n}. Let S_{n,k} be the join of a complete graph of order k and an independent set of order n-k. It is shown that if G is a graph of order n, with no 5-cycle, then q(G)<q(S_{n,2}), unless G=S_{n,k}. It is shown that these results are significant in spectral extremal graph problems. Two conjectures are formulated for the maximum q(G) of graphs with forbidden cycles.Comment: 12 page

Possible Influence of Natural Events on Heavy Metals Exposure from Shellfish Consumption: A Case Study in the North-East of Italy

The objective of this study was the estimation of the exposure over time to heavy metals (cadmium, mercury and lead) due to shellfish consumption in the Veneto Region, Italy. Shellfish consumption was investigated by a food frequency consumption survey. Altogether, 1949 households, stratified into the five most populated areas of the Veneto Region, were involved in the study. Exposure estimation to heavy metals was carried out taking into account the level of metal measured in samples of Manila clams (Ruditapes philippinarum) and grooved carpetshell (Ruditapes decussatus), collected in the frame of the monitoring activities of mollusk production areas of Veneto Region, between January 2007 and December 2012. A general high contribution of the considered shellfish to the Tolerable Weekly Intake (TWI) was noticed in the case of cadmium, especially in 2011, when a considerable increase in cadmium intake was estimated. This was probably due to a heavy rainfall event that triggered catastrophic flooding with high impact on shellfish capture areas in November 2010. The results strongly emphasise the importance of dealing with food safety in a holistic way, taking into account the potential impact of extraordinary natural events on food chain contamination, in order to identify food hazards at an early stage, before developing into a real risk for consumers

Sobre vértices do esqueleto do politopo de emparelhamentos de um grafo

O politopo de emparelhamentos de um grafo G, M(G), e ́ o fecho convexo dos vetores de incidência de emparelhamentos de G. O esqueleto deste politopo, G(M(G)), e ́ o grafo cujos vértices e arestas são, respectivamente, os vértices e arestas de M(G). Neste trabalho calculamos o grau do vértice do esqueleto correspondente ao emparelhamento vazio. Mostramos que, dado qualquer subgrafo próprio H de um grafo G, o grau de um vértice de G(M(H)) e ́ estritamente menor que o grau deste em G(M(G)). Além disso, determinamos o número de vértices e o grau mínimo (e máximo, em alguns casos) do esqueleto do politopo de emparelhamentos de grafos pertencentes a duas classes: a primeira, constituída por grafos unicíclicos obtidos pela adição de uma aresta entre dois vértices não adjacentes de um caminho; a segunda, dada por grafos resultantes da ligação de um dado vértice a todos os vértices de uma estrela

Maxima of the Q-index: Graphs with no Ks,t

This note presents a new spectral version of the graph Zarankiewicz problem: How large can be the maximum eigenvalue of the signless Laplacian of a graph of order n that does not contain a specified complete bipartite subgraph. A conjecture is stated about general complete bipartite graphs, which is proved for infinitely many cases. More precisely, it is shown that if G is a graph of order n, with no subgraph isomorphic to K2,s+1, then the largest eigenvalue q(G) of the signless Laplacian of G satisfiesq(G) (Formula presented.), with equality holding if and only if G is a join of K1 and an s-regular graph of order n-1

Maxima of the Q-index: Forbidden 4-cycle and 5-cycle

This paper gives tight upper bounds on the largest eigenvalue q (G) of the signless Laplacian of graphs with no 4-cycle and no 5-cycle. If n is odd, let Fn be the friendship graph of order n; if n is even, let Fn be Fn-1 with an extra edge hung to its center. It is shown that if G is a graph of order n ≥ 4, with no 4-cycle, then q (G) \u3c q (Fn), unless G = Fn. Let Sn,k be the join of a complete graph of order k and an independent set of order n - k. It is shown that if G is a graph of order n ≥ 6, with no 5-cycle, then q (G) \u3c q (Sn,2), unless G = Sn,k. It is shown that these results are significant in spectral extremal graph problems. Two conjectures are formulated for the maximum q (G) of graphs with forbidden cycles

Testing nano-silver food packaging to evaluate silver migration and food spoilage bacteria on chicken meat

ABSTRACT Migration of nanomaterials from food containers into food is a matter of concern because of the potential risk for exposed consumers. The aims of this study were to evaluate silver migration from a commercially available food packaging containing silver nanoparticles into a real food matrix (chicken meat) under plausible domestic storage conditions and to test the contribution of such packaging to limit food spoilage bacteria proliferation. Chemical analysis revealed the absence of silver in chicken meatballs under the experimental conditions in compliance with current European Union legislation, which establishes a maximum level of 0.010 mg kg\u20131 for the migration of non-authorised substances through a functional barrier (Commission Regulation (EU) No. 10/2011). On the other hand, microbiological tests (total microbial count, Pseudomonas spp. and Enterobacteriaceae) showed no relevant difference in the tested bacteria levels between meatballs stored in silver-nanoparticle plastic bags or control bags. This study shows the impor- tance of testing food packaging not only to verify potential silver migration as an indicator of potential nanoparticle migration, but also to evaluate the benefits in terms of food preservation so as to avoid unjustified usage of silver nanoparticles and possible negative impacts on the environment