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Restricted star colouring is a variant of star colouring introduced to design heuristic algorithms to estimate sparse Hessian matrices. For k ∈ N , a k -restricted star colouring ( k -rs colouring) of a graph G is a function f : V ( G ) → { 0 , 1 , … , k − 1 } such that (i) f ( x ) ≠ f ( y ) for every edge x y of G , and (ii) there is no bicoloured 3-vertex path ( P 3 ) in G with the higher colour on its middle vertex. We show that for k ≥ 3 , it is NP-complete to test whether a given planar bipartite graph of maximum degree k and arbitrarily large girth admits a k -rs colouring, and thereby answer a problem posed by Shalu and Sandhya (2016). In addition, it is NP-complete to test whether a 3-star colourable graph admits a 3-rs colouring. We also prove that for all ϵ > 0 , the optimization problem of restricted star colouring a 2-degenerate bipartite graph with the minimum number of colours is NP-hard to approximate within n 1 3 − ϵ . On the positive side, we design (i) a linear-time algorithm to test 3-rs colourability of trees, and (ii) an O ( n 3 ) -time algorithm to test 3-rs colourability of chordal graphs.

A $(k,g)$-cage is a $k$-regular simple graph of girth $g$ with minimum possible number of vertices. In this paper, $(k,g)$-cages which are Moore graphs are referred as minimal $(k,g)$-cages. A simple connected graph is called distance regular(DR) if all its vertices have the same intersection array. A bipartite graph is called distance biregular(DBR) if all the vertices of the same partite set admit the same intersection array. It is known that minimal $(k,g)$-cages are DR graphs and their subdivisions are DBR graphs. In this paper, for minimal $(k,g)$-cages we give a formula for distance spectral radius in terms of $k$ and $g$, and also determine polynomials of degree $[\frac{g}{2}]$, which is the diameter of the graph. This polynomial gives all distance eigenvalues when the variable is substituted by adjacency eigenvalues. We show that a minimal $(k,g)$-cage of diameter $d$ has $d+1$ distinct distance eigenvalues, and this partially answers a problem posed in [5]. We prove that every DBR graph is a $2$-partitioned transmission regular graph and then give a formula for its distance spectral radius. By this formula we obtain the distance spectral radius of subdivision of minimal $(k,g)$-cages. Finally we determine the full distance spectrum of subdivision of some minimal $(k,g)$-cages., 22 pages

Hibernation, a metabolic strategy, allows individuals to reduce energetic demands in times of energetic deficits. Hibernation is pervasive in nature, occurring in all major mammalian lineages and geographical regions; however, its expression is variable across species, populations, and individuals, suggesting that trade-offs are at play. Whereas hibernation reduces energy expenditure, energetically expensive arousals may impose physiological burdens. The torpor optimization hypothesis posits that hibernation should be expressed according to energy availability. The greater the energy surplus, the lower the hibernation output. The thrifty female hypothesis, a variation of the torpor optimization hypothesis, states that females should conserve more energy because of their more substantial reproductive costs. Contrarily, if hibernation's benefits offset its costs, hibernation may be maximized rather than optimized (e.g., hibernators with greater fat reserves could afford to hibernate longer). We assessed torpor expression in captive dwarf lemurs, primates that are obligate, seasonal, and tropical hibernators. Across 4.5 mo in winter, we subjected eight individuals at the Duke Lemur Center to conditions conducive to hibernation, recorded estimates of skin temperature hourly (a proxy for torpor), and determined body mass and tail fat reserves bimonthly. Across and between consecutive weigh-ins, heavier dwarf lemurs spent less time in torpor and lost more body mass. At equivalent body mass, females spent more time torpid and better conserved energy than did males. Although preliminary, our results support the torpor optimization and thrifty female hypotheses, suggesting that individuals optimize rather than maximize torpor according to body mass. These patterns are consistent with hibernation phenology in Madagascar, where dwarf lemurs hibernate longer in more seasonal habitats.

We propose a new model of asynchronous batch codes that allow for parallel recovery of information symbols from a coded database in an asynchronous manner, i.e. when queries arrive at random times and they take varying time to process. We show that the graph-based batch codes studied by et al. are asynchronous. Further, we demonstrate that hypergraphs of Berge girth larger or equal to 4, respectively larger or equal to 3, yield graph-based asynchronous batch codes, respectively private information retrieval (PIR) codes. We prove the hypergraph-theoretic proposition that the maximum number of hyperedges in a hypergraph of a fixed Berge girth equals the quantity in a certain generalization of the hypergraph-theoretic (6,3)-problem, first posed by Brown, Erd\H{o}s and S\'{o}s. We then apply the constructions and bounds by Erd\H{o}s, Frankl and R\"{o}dl about this generalization of the (6,3)-problem, known as the (3$\varrho$-3,$\varrho$)-problem, to obtain batch code constructions and bounds on the redundancy of the graph-based asynchronous batch and PIR codes. We derive bounds on the optimal redundancy of several families of asynchronous batch codes with the query size $t=2$. In particular, we show that the optimal redundancy $\rho(k)$ of graph-based asynchronous batch codes of dimension $k$ for $t=2$ is $2\sqrt{k}$. Moreover, for graph-based asynchronous batch codes with $t \ge 3$, $\rho(k) = O\left({k}^{1/(2-\epsilon)}\right)$ for any small $\epsilon>0$., Comment: 18 pages

A b-coloring of a graph is a proper coloring of its vertices such that each color class contains a vertex adjacent to at least one vertex of every other color class. The b-chromatic number of a graph is the largest integer k such that the graph has a b-coloring with k colors. In this work we present lower bounds for the b-chromatic number of a vertex-deleted subgraph of a graph, particularly regarding two important classes, claw-free graphs and chordal graphs. We also get bounds for the b-chromatic number of G − { x } , when G is a graph with large girth.

The Grundy and b-chromatic number of graphs are two important chromatic parameters. The Grundy number of a graph G , denoted by Γ ( G ) is the worst case behavior of greedy (First-Fit) coloring procedure for G and the b-chromatic number b ( G ) is the maximum number of colors used in any color-dominating coloring of G . Because the nature of these colorings are different they have been studied widely but separately in the literature. In this paper we first prove that Γ ( G ) − ⌈ log Γ ( G ) ⌉ ≤ b ( G ) , if the girth of G is sufficiently large with respect to its maximum degree. Next, we prove that if G is K 2 , 3 -free then Γ ( G ) ≤ ( b ( G ) ) 3 / 2 . These results confirm a previous conjecture for these families of graphs.

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.

In this paper, we introduce a graph topological structure called Inclusion Graph of a Topology . The diameter, girth, connectivity, maximal independent sets, different variants of domination number...

We prove that the ‘Upper Matching Conjecture’ of Friedland, Krop, and Markstrom and the analogous conjecture of Kahn for independent sets in regular graphs hold for all large enough graphs as a function of the degree. That is, for every d and every large enough n divisible by 2d, a union of n / ( 2 d ) copies of the complete d -regular bipartite graph maximizes the number of independent sets and matchings of size k for each k over all d-regular graphs on n vertices. To prove this we utilize the cluster expansion for the canonical ensemble of a statistical physics spin model, and we give some further applications of this method to maximizing and minimizing the number of independent sets and matchings of a given size in regular graphs of a given minimum girth.

Lin-Lu-Yau introduced a notion of Ricci curvature for graphs and obtained a complete classification for all Ricci-flat graphs with girth at least five. In this paper, we characterize all Ricci-flat graphs of girth four with vertex-disjoint 4-cycles.

A subset of vertices in a graph is called a total dominating set if every vertex of the graph is adjacent to at least one vertex of this set. A total dominating set is called minimal if it does not properly contain another total dominating set. In this paper, we study graphs whose all minimal total dominating sets have the same size, referred to as well-totally-dominated (WTD) graphs. We first show that WTD graphs with bounded total domination number can be recognized in polynomial time. Then we focus on WTD graphs with total domination number two. In this case, we characterize triangle-free WTD graphs and WTD graphs with packing number two, and we show that there are only finitely many planar WTD graphs with minimum degree at least three. Lastly, we show that if the minimum degree is at least three then the girth of a WTD graph is at most 12. We conclude with several open questions., Comment: 13 pages, 2 figures

A variation of the Cops and Robber game is introduced in which the robber side consists of two robbers. The cops win by moving onto the same vertex as one of the robbers after a finite number of moves. As in the original game, the robber side can win by avoiding capture indefinitely. In this version, however, the robbers can also win by both moving onto the same vertex as the cop at the same time. Otherwise, the robbers must be located on distinct vertices. We present structural properties of ambush-copwin graphs, those graphs on which a single cop can guarantee a win. As well, we characterize ambush-copwin graphs of girth $$g \ge 4$$ and classes of ambush-copwin graphs of girth 3, particularly a class of chordal graphs.

In this paper, we first prove that the connected cubic edge-transitive bi-Cayley graphs over a generalized dihedral group have girth 6. Using this, a complete classification is given of these graphs.

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity a v e c ( G ) of a graph G is the mean value of eccentricities of all vertices of G . In this paper, we introduce some transformations for connected graphs that do not decrease their average eccentricity. Using these transformations, we completely determine the graphs which have the maximum average eccentricity among all connected graphs of order n with girth g and maximum degree Δ . Moreover, the graphs with maximum average eccentricity in some classes of graphs are characterized.

A graph is called equimatchable if all of its maximal matchings have the same size. Frendrup et al. [8] provided a characterization of equimatchable graphs with girth at least $5$. In this paper, we extend this result by providing a complete structural characterization of equimatchable graphs with girth at least $4$, i.e., equimatchable graphs with no triangle, by identifying the equimatchable triangle-free graph families. Our characterization also extends the result given by Akbari et al. in [1], which proves that the only connected triangle-free equimatchable $r$-regular graphs are $C_5$, $C_7$ and $K_{r,r}$, where $r$ is a positive integer. Given a non-bipartite graph, our characterization implies a linear time recognition algorithm for triangle-free equimatchable graphs.

Mangrove species, viz., Avicennia officinalis, Avicennia marina var. acutissima, Avicennia marina (dwarf), Rhizophora mucronata, Sonneratia alba, Aegiceras corniculatum, Kandelia candel were chosen for measurement of height of the tree and girth or circumference. The sampling was random and at least 50 records were made. The girth is measured by the tape. The measurement of the height is made with the help of abny level. The Tables 1 to 8 records the values for girth, height and canopy cover as well as for correlation coefficient (r). There correlation between girth and canopy in all the species studied however in case of Avicennia officinalis and Aegiceras corniculatum girth and height show more co-relation than girth and C. cover. The positive co-relation observed between girth and canopy is more or less 0.7 except Avicennia marina (dwarf) Excoecaria agallocha and Aegiceras corniculatum. The co-relation is observed in girth and height is difficult to explain. This case is observed in Avicennia officinalis and Aegiceras corniculatum.

The greatest-common-divisor (GCD) method is a general framework employing a set of simple inequalities (called GCD constraint) to guarantee girth eight for a class of $(J,L)$ quasi-cyclic (QC) low-density parity-check (LDPC) codes. However, an important problem, i.e., whether the GCD constraint is necessary for this class of codes to have girth eight, remains open. In this letter, the question is answered affirmatively, following which a novel algorithm aiming to find the shortest codes with girth eight in such a class is proposed. Besides, a close connection is established between the GCD method and the base expansion method, which are both applicable for any $J$ and any $L$ .

G has a list k-L(2, 1)-labeling if for any k-list assignment L, there exists a coloring $$c:V(G)\rightarrow \bigcup \limits _{v\in V} L(v)$$ of G such that $$c(v)\in L(v)$$ for $$\forall v\in V(G)$$ and for $$\forall u,v\in V(G)$$ , $$|c(u)-c(v)|\ge 2$$ if $$d(u,v)=1$$ , $$|c(u)-c(v)|\ge 1$$ if $$d(u,v)=2$$ . $$\lambda _{2,1}^{l}(G)=\min \{k|G$$ has a list k-L(2, 1)-labeling $$\}$$ is called the list L(2, 1)-labeling number of G. In this paper, we prove that for planar graph with maximum degree $$\Delta \ge 5$$ , girth $$g\ge 13$$ and without adjacent 13-cycles, $$\lambda _{2,1}^{l}(G)\le \Delta +3$$ holds. Moreover, the upper bound $$\Delta +3$$ is tight.

Let $ G $ be a simple graph with edge set $ E(G) $. The modified Sombor index is defined as $ ^{m}SO(G) = \sum\limits_{uv\in E(G)}\frac{1}{\sqrt{d_{u}^{2}~~+~~d_{v}^{2}}} $, where $ d_{u} $ (resp. $ d_{v} $) denotes the degree of vertex $ u $ (resp. $ v $). In this paper, we determine some bounds for the modified Sombor indices of graphs with given some parameters (e.g., maximum degree $ \Delta $, minimum degree $ \delta $, diameter $ d $, girth $ g $) and the Nordhaus-Gaddum-type results. We also obtain the relationship between modified Sombor index and some other indices. At last, we obtain some bounds for the modified spectral radius and energy.

We introduce a new approach and prove that the maximum number of triangles in a $C_5$-free graph on $n$ vertices is at most $$(1 + o(1)) \frac{1}{3 \sqrt 2} n^{3/2}.$$ We also show a connection to $r$-uniform hypergraphs without (Berge) cycles of length less than six, and estimate their maximum possible size., Comment: 14 pages

Bermond et al in 1980 proved that if the size of a strong digraph D of order n is at least $$\left( {\begin{array}{c}n-k+2\\ 2\end{array}}\right) +k-1$$ , $$k\ge 2$$ , then the girth of D is no more than k. Consequently, when D is a 3-free strong digraph of order n without loops or parallel arcs, which means that every directed cycle in D has length at least 4, the maximum size of D is $$\left( {\begin{array}{c}n-1\\ 2\end{array}}\right) +1$$ . In 2008, Seymour et al proved that if D is a 3-free digraph, then $$\beta (D)\le \gamma (D)$$ , and they further conjectured that $$\beta (D)\le \frac{1}{2}\gamma (D)$$ for every 3-free digraph D, where $$\beta (D)$$ denotes the size of the smallest subset $$X\subseteq A(D)$$ , such that $$D\backslash X$$ is acyclic, and $$\gamma (D)$$ is the number of unordered pairs $$\{u,v\}$$ of vertices such that u, v are nonadjacent in D. In this paper, we first describe all 3-free strong digraphs of order n with the maximum size $$\left( {\begin{array}{c}n-1\\ 2\end{array}}\right) +1$$ . Then we prove that such 3-free strong digraphs satisfy the minimum out-degree of D equals 1 and $$\beta (D)\le \frac{1}{2}\gamma (D)$$ . Moreover, we prove that if the minimum out-degree of a 3-free strong digraph D is at least 2, then the maximum size of D is between $$\left( {\begin{array}{c}n-1\\ 2\end{array}}\right) -2$$ and $$\left( {\begin{array}{c}n-1\\ 2\end{array}}\right)$$ .

The present study aimed to describe and critically discuss the current evidence regarding the penile girth enhancement procedures for aesthetic purposes. We designed a narrative review of the literature. A comprehensive search in the MEDLINE database was performed. Original articles in English-language, published until March 2021, were selected. A total of 29 studies were included (3 reporting non-invasive approaches, 11 injection therapies, and 15 surgical procedures). The vast majority of articles (26, 89.7%) were not randomized controlled trials, with overall low quality and limited level of evidence. Only 1 (33.3%) paper regarding non-invasive approaches reported a minimal (+0.03 cm) but a significant increase of penile girth (p = 0.034). A low rate (11.2-14.4%) of mild, temporary adverse events and poor-to-moderate patient satisfaction were found. Eight (72.7%) articles concerning injection therapies showed a significant increase in penile girth (p < 0.05). A low rate of mild complications, generally at the injection site, and a high patient satisfaction rate (75-100%) were highlighted. Nine (60%) papers on surgical treatments found a significant increase in penile girth (p < 0.05), while the other 6 (40%) studies reported a generic improvement in penile circumference. Skin necrosis or ulcers, wound infections, or need for reoperation were reported in 8 (53.3%) studies. A high patient satisfaction rate (60-100%) was reported. Our review highlighted the overall positive results of injection procedures, the poor outcomes associated with non-invasive techniques, and the good efficacy and satisfaction with a non-negligible risk of complications in patients undergoing surgical treatments. However, the adverse events are probably largely under-reported and these procedures should still be considered under investigation due to the limited evidence available and the lack of guidelines.

An edge-girth-regular $$egr(v,k,g,\lambda )$$ -graph $$\Gamma $$ is a k-regular graph of order v and girth g in which every edge is contained in $$\lambda $$ distinct g-cycles. Edge-girth-regularity is shared by several interesting classes of graphs which include edge- and arc-transitive graphs, Moore graphs, as well as many of the extremal k-regular graphs of prescribed girth or diameter. Infinitely many $$egr(v,k,g,\lambda )$$ -graphs are known to exist for sufficiently large parameters $$(k,g,\lambda )$$ , and in line with the well-known Cage Problem we attempt to determine the smallest graphs among all edge-girth-regular graphs for given parameters $$(k,g,\lambda )$$ . To facilitate the search for $$egr(v,k,g,\lambda )$$ -graphs of the smallest possible orders, we derive lower bounds in terms of the parameters k, g and $$\lambda $$ . We also determine the orders of the smallest $$egr(v,k,g,\lambda )$$ -graphs for some specific parameters $$(k,g,\lambda )$$ , and address the problem of the smallest possible orders of bipartite edge-girth-regular graphs.

In this article, we introduce and study the intersection graph of graded ideals of a graded ring. The intersection graph of $ G- $graded ideals of a graded ring $ R $, denoted by $ Gr_G(R) $, is undirected simple graph defined on the set of nontrivial graded left ideals of $ R $, such that two left ideals are adjacent if their intersection is not trivial. We study properties for these graphs such as connectivity, regularity, completeness, domination numbers, and girth. We also present several results on the intersection graphs related to faithful grading, strong grading, and graded idealization.

Through failure generalization, fracture feature analysis and material performance test, a comprehensively analysis was made on the fracture failure analysis for girth weld of gathering pipelines containing H2S gas. The results showed that the fracture failure might be mainly due to sulfide stress cracking in the girth weld. The crack originated from the fusion line on the inner surface of girth weld and extended along the girth weld to outside closed to bends. The sulfide stress cracking of the girth weld was caused by the intersection of multiple factors. The service condition was located in SSC 3 zone and the SSC risk of girth weld was high. The girth weld itself was not been stress-relieved, and its ability to resist SSC was poor. Due to low wall thickness, welding defects, welding stress and additional load, the actual stress of weld was higher.

We study the graph of zero-divisors of a quadratic quotient of a Rees algebra. We provide conditions in order to determine its girth and diameter, extending several results regarding amalgamated du...

In this paper, we introduce a graph topological structure, called Intersection Graph Iτ(X) = (V(τ), E(τ)), where V(τ) = Collection of all topologies defined on X other than τi and τd and for τ1, τ2...

In this paper, we introduce and study a new distance parameter {\it triameter} of a connected graph $G$, which is defined as $max\{d(u,v)+d(v,w)+d(u,w): u,v,w \in V\}$ and is denoted by $tr(G)$. We find various upper and lower bounds on $tr(G)$ in terms of order, girth, domination parameters etc., and characterize the graphs attaining those bounds. In the process, we provide some lower bounds of (connected, total) domination numbers of a connected graph in terms of its triameter. The lower bound on total domination number was proved earlier by Henning and Yeo. We provide a shorter proof of that. Moreover, we prove Nordhaus-Gaddum type bounds on $tr(G)$ and find $tr(G)$ for some specific family of graphs., Comment: 16 pages

In this paper we present bounds on the radius and diameter of graphs of girth at least 6 and for ( C 4 , C 5 ) -free graphs, i.e., graphs not containing cycles of length 4 or 5. We show that the diameter of a graph G of girth at least 6 is at most 3 n δ 2 − δ + 1 − 1 , and the radius is at most 3 n 2 ( δ 2 − δ + 1 ) + 10 , where n is the order and δ the minimum degree of G . If δ − 1 is a prime power, then both bounds are sharp apart from an additive constant. For graphs of large maximum degree Δ , we show that these bounds can be improved to 3 n − Δ δ δ 2 − δ + 1 − 3 ( δ − 1 ) Δ ( δ − 2 ) δ 2 − δ + 1 + 10 for the diameter, and 3 n − 3 Δ δ 2 ( δ 2 − δ + 1 ) − 3 ( δ − 1 ) Δ ( δ − 2 ) 2 ( δ 2 − δ + 1 ) + 22 for the radius. We further show that only slightly weaker bounds hold for ( C 4 , C 5 ) -free graphs. As a by-product we obtain a result on a generalisation of cages. For given δ , Δ ∈ N with Δ ≥ δ let n ( δ , Δ , g ) be the minimum order of a graph of girth g , minimum degree δ and maximum degree Δ . Then n ( δ , Δ , 6 ) ≥ Δ δ + ( δ − 1 ) Δ ( δ − 2 ) + 3 2 . If δ − 1 is a prime power, then there exist infinitely many values of Δ such that, for δ constant and Δ large, n ( δ , Δ , 6 ) = δ Δ + O ( Δ ) .

Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The weakly zero-divisor graph of R is the undirected (simple) graph WΓ(R) with vertex set Z(R) ∗, and two distinct vertices x and y are adjacent if and only if there exist r ∈ ann(x) and s ∈ ann(y) such that rs = 0. It follows that WΓ(R) contains the zero-divisor graph Γ(R) as a subgraph. In this paper, the connectedness, diameter, and girth of WΓ(R) are investigated. Moreover, we determine all rings whose weakly zero-divisor graphs are star. We also give conditions under which weakly zero-divisor and zero-divisor graphs are identical. Finally, the chromatic number of WΓ(R) is studied.

Let $\mathcal C_{n}$ be the Catalan monoid on $X_{n}=\{1,\ldots ,n\}$ under its natural order. In this paper, we describe the sets of left zero-divisors, right zero-divisors and two sided zero-divisors of $\mathcal C_{n}$; and their numbers. For $n \geq 4$, we define an undirected graph $\Gamma(\mathcal C_{n})$ associated with $\mathcal C_{n}$ whose vertices are the two sided zero-divisors of $\mathcal C_{n}$ excluding the zero element $\theta$ of $\mathcal C_{n}$ with distinct two vertices $\alpha$ and $\beta$ joined by an edge in case $\alpha\beta=\theta=\beta\alpha$. Then we first prove that $\Gamma(\mathcal C_{n})$ is a connected graph, and then we find the diameter, radius, girth, domination number, clique number and chromatic numbers and the degrees of all vertices of $\Gamma(\mathcal C_{n})$. Moreover, we prove that $\Gamma(\mathcal C_{n})$ is a chordal graph, and so a perfect graph.

Time-varying spatially coupled low-density parity-check (SC-LDPC) codes with very large period are characterized by significantly better error rate performance and girth properties than their time-invariant counterparts, but the number of parameters they require to be described is usually very large and unpractical. Time-invariant SC-LDPC codes, which can be seen as periodically time-varying codes with unitary period, are represented through a small number of parameters and designed exploiting few degrees of freedom, but their error rate performance and girth properties are sub-optimal. In this paper, we show that the limits of time-invariant SC-LDPC codes can be overcome by transforming them into time-varying SC-LDPC codes with very small period. In particular, we show that periodically time-varying SC-LDPC codes with small period may exhibit significantly better girth properties than the corresponding time-invariant codes by exploiting a larger number of degrees of freedom in the code design, which however scale at most linearly with the product of the code period and the size of the considered base matrix.

In this paper, two Brualdi-type eigenvalue inclusion sets are presented, which are characterized by the girth of the digraph associated with a tensor and are proved weaker than the conditions of Brualdi-type eigenvalue inclusion sets given by Bu et al. (2015) [7] . As applications of the above results, the conditions of nonsingularity and positive definiteness of tensors are given.

In this paper we study group actions on quasi-median graphs, or 'CAT(0) prism complexes', generalising the notion of CAT(0) cube complexes. We consider hyperplanes in a quasi-median graph $X$ and define the contact graph $\mathcal{C}X$ for these hyperplanes. We show that $\mathcal{C}X$ is always quasi-isometric to a tree, generalising a result of Hagen, and that under certain conditions a group action $G \curvearrowright X$ induces an acylindrical action $G \curvearrowright \mathcal{C}X$, giving a quasi-median analogue of a result of Behrstock, Hagen and Sisto. As an application, we exhibit an acylindrical action of a graph product on a quasi-tree, generalising results of Kim and Koberda for right-angled Artin groups. We show that for many graph products $G$, the action we exhibit is the 'largest' acylindrical action of $G$ on a hyperbolic metric space. We use this to show that the graph products of equationally noetherian groups over finite graphs of girth $\geq 6$ are equationally noetherian, generalising a result of Sela., 36 pages, 11 figures; a minor revision. To appear in Groups, Geometry and Dynamics

Spacapan recently showed that there exist 3-polytopes with non-Hamiltonian prisms, disproving a conjecture of Rosenfeld and Barnette. By adapting Spacapan's approach we strengthen his result in several directions. We prove that there exists an infinite family of counterexamples to the Rosenfeld-Barnette conjecture, each member of which has maximum degree 37, is of girth 4, and contains no odd-length face with length less than k for a given odd integer k. We also show that for any given 3-polytope H there is a counterexample containing H as an induced subgraph. This yields an infinite family of non-Hamiltonian 4-polytopes in which the proportion of quartic vertices tends to 1. However, Barnette's conjecture stating that every 4-polytope in which all vertices are quartic is Hamiltonian still stands. Finally, we prove that the Grunbaum-Walther shortness coefficient of the family of all prisms of 3-polytopes is at most 59/60.

In order to eliminate the salt pepper and Gaussian mixed noise in X-ray weld image, the extreme value characteristics of salt and pepper noise are used to separate the mixed noise, and the non local mean filtering algorithm is used to denoise it. Because the smoothness of the exponential weighted kernel function is too large, it is easy to cause the image details fuzzy, so the cosine coefficient based on the function is adopted. An improved non local mean image denoising algorithm is designed by using weighted Gaussian kernel function. The experimental results show that the new algorithm reduces the noise and retains the details of the original image, and the peak signal-to-noise ratio is increased by 1.5 dB. An adaptive salt and pepper noise elimination algorithm is proposed, which can automatically adjust the filtering window to identify the noise probability. Firstly, the median filter is applied to the image, and the filtering results are compared with the pre filtering results to get the noise points. Then the weighted average of the middle three groups of data under each filtering window is used to estimate the image noise probability. Before filtering, the obvious noise points are removed by threshold method, and then the central pixel is estimated by the reciprocal square of the distance from the center pixel of the window. Finally, according to Takagi Sugeno (T-S) fuzzy rules, the output estimates of different models are fused by using noise probability. Experimental results show that the algorithm has the ability of automatic noise estimation and adaptive window adjustment. After filtering, the standard mean square deviation can be reduced by more than 20%, and the speed can be increased more than twice. In the enhancement part, a nonlinear image enhancement method is proposed, which can adjust the parameters adaptively and enhance the weld area automatically instead of the background area. The enhancement effect achieves the best personal visual effect. Compared with the traditional method, the enhancement effect is better and more in line with the needs of industrial field.

PurposeThe authors study the interdisciplinary relation between graph and algebraic structure ring defining a new graph, namely “non-essential sum graph”. The nonessential sum graph, denoted by NES(R), of a commutative ringRwith unity is an undirected graph whose vertex set is the collection of all nonessential ideals ofRand any two vertices are adjacent if and only if their sum is also a nonessential ideal ofR.Design/methodology/approachThe method is theoretical.FindingsThe authors obtain some properties of NES(R) related with connectedness, diameter, girth, completeness, cut vertex,r-partition and regular character. The clique number, independence number and domination number of NES(R) are also found.Originality/valueThe paper is original.

In this paper, we consider a randomized greedy algorithm for independent sets in $r$-uniform $d$-regular hypergraphs $G$ on $n$ vertices with girth $g$. By analyzing the expected size of the independent sets generated by this algorithm, we show that $\alpha(G)\geq (f(d,r)-\epsilon(g,d,r))n$, where $\epsilon(g,d,r)$ converges to $0$ as $g\rightarrow\infty$ for fixed $d$ and $r$, and $f(d,r)$ is determined by a differential equation. This extends earlier results of Gamarnik and Goldberg for graphs. We also prove that when applying this algorithm to uniform linear hypergraphs with bounded degree, the size of the independent sets generated by this algorithm concentrate around the mean asymptotically almost surely.

For every positive integer $n$, we construct a Hasse diagram with $n$ vertices and chromatic number $\Omega(n^{1/4})$, which significantly improves on the previously known best constructions of Hasse diagrams having chromatic number $\Theta(\log n)$. In addition, if we also require that our Hasse diagram has girth at least $k\geq 5$, we can achieve a chromatic number of at least $n^{\frac{1}{2k-3}+o(1)}$. These results have the following surprising geometric consequence. They imply the existence of a family $\mathcal{C}$ of $n$ curves in the plane such that the disjointness graph $G$ of $\mathcal{C}$ is triangle-free (or have high girth), but the chromatic number of $G$ is polynomial in $n$. Again, the previously known best construction, due to Pach, Tardos and T\'oth, had only logarithmic chromatic number., Comment: 11 pages, 1 figure

Linear and geometric regression equations were used to estimate liveweight of two hundred and eighty six semi-intensively managed West African Dwarf (WAD) goats from Chest girth (CG) and Wither height (WH) measurements. CG accounted for 52 and 58% variation in the liveweight of goats using linear and geometric equations respectively, thus showing a curvilinear relationship. WH accounted for 31% variation in liveweight of goats using both equations. CG measurement was found to be more reliable for estimating liveweight of WAD goats than WH measurement. Using linear regression equation, CG and WH estimated the liveweight of females better than males with average increases of 57% and 25% R' values respectively. Using geometric regression equation, CG and WH also estimated the liveweight of females better than males with average increases of 55% and 22% R' values respectively. In essence, CG and WH measurements estimated the liveweighi of does better than bucks using both regression equations. Generally, CG and WH better estimated the liveweight of goats that are greater or equal to 3 years old than goats that are greater than / year but less than 3 years old and goats that are less or equal to 1 year old using both regression equations. Even though, each of the geometric or linear regression equations may be used to estimate liveweight of WAD goats with a high degree of reliability in a specific class of animals, it would be simpler and less cumbersome to use the regression equation obtained from pooled data to estimate liveweight in all animals

The pre weaning growth performance of piglets from four different cross bred dams mated back to their native parents was studied in the Breeding and Research Farm of Ebonyi State University, Abakaliki. Parameters like body weight, weight gain, feed intake and feed efficiency as well as linear body measurements of body length, heart girth and height at withers were taken and observed for growth between birth and weaning age of eight weeks. Results Results indicated significant differences (P

In this paper, we determine automorphisms of j(τ) and prove some results on j(τ),when topological space (X, τ) is finite. Also we determine necessary and sufficient condition for the open subset in...

The prediction of pig liveweight from three body measurements was studied using crossbred (Landrace x Large White and Hampshire x Large White) barrows and gilts, there were 286 measurements for the different parameters studied (Chest girth, Body length and Lain girth). "There were no significant differences due to sex for all the parameters investigated. Pig liveweight was best predicted singly using chest girth measurement. However, with the use of a combination of chest girth and body length, the prediction efficiency was enhanced.

Let G = (V, E) be any connected graph with for all uj, uk ∈ Si if e(uj) = e(uk)(1 ≤ i ≤ t) with each | Si|≥ 2 and . The eccentricity splitting graph of a graph denoted by ES(G) is obtained by takin...

Multitype quasi-cyclic (QC) low-density parity-check (LDPC) codes are a class of protograph LDPC codes lifted cyclically from protographs with multiple edges, represented by two weight and slope matrices. For a given weight-matrix, an approach is proposed to find the maximum-achievable girth $g_{\max }$ of the corresponding multitype QC-LDPC codes by some inevitable chains having less complexity than the existing methods. This advantage leads to some new patterns of the weight matrices such that the corresponding codes have some improvements in terms of the maximum-achievable girths or the minimum-distance upper-bounds. In continue, for a given weight-matrix with maximum-achievable girth $g_{\max }$ , some slope-matrices are constructed by a depth-first search algorithm for which the corresponding multitype QC-LDPC codes with even girth $g$ , $g\le g_{\max }$ , have smaller lengths, higher rates, or larger minimum-distances than the state-of-the-art achievements. Simulation results show that the constructed codes have some error-rate advantages than PEG, random-like, CCSDS, and 802.11n/ac IEEE standard LDPC codes.

In this article, we design finite-length irregular protograph-based quasi-cyclic (QC) low-density parity-check (LDPC) codes with good waterfall performance and low error floor. To achieve a low error floor, we eliminate a targeted set of dominant elementary trapping sets (ETS) ${\mathcal{ L}}$ in the Tanner graph of the code. For a given rate and girth, the codes are designed to be free of the largest set of problematic ETSs for a given block length, or to have the shortest block length while a given set of ETSs is avoided. The design is based on a search algorithm that identifies whether any instance of any structure within ${\mathcal{ L}}$ exists in the Tanner graph of the constructed code or not. The search algorithm performs this task with minimal complexity, making it feasible to construct practical codes by running the search algorithm a large number of times. Simulation results are provided to demonstrate the superior performance of designed codes compared to similar state-of-the-art irregular QC-LDPC codes.

In this article, we study the cycle structure of (3, 19)-regular Tanner’s quasi-cyclic (QC) LDPC codes with code length $19p$ , where $p$ is a prime and $p\equiv 1~(\bmod ~57)$ , and transform the conditions for the existence of cycles of lengths not more than 10 into polynomial equations in a 57th root of unity of the prime field $\mathbb {F}_{p}$ . By employing the Euclidean division algorithm to check whether these equations have solutions over the prime field $\mathbb {F}_{p}$ , the girth values of (3, 19)-regular Tanner’s QC-LDPC codes of code length $19p$ are determined. In order to show the good performance of this class of QC-LDPC codes, numerical results are also provided.