# A Class of Koszul Algebra and Some Homological Invariants through Circulant Matrices and Cycles

Authors: Muhammed Nadeem; Mohammed M. M. Jaradat; Imran Siddique; Muhammad Arfan Ali; Muhammad Ahsan; Muhammad Sarwar Sindhu
Journal: Journal of Mathematics
 (2022)

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## Abstract

Recent advances in graph theory, linear algebra, and commutative algebra render us to tackle problems in one bough of mathematics with assistance and guidance from others. We will elaborate foremost and conceptually fathomless homological invariants inextricably linked with circulant matrices and cycles through various path lengths in this article, as well as a class of Koszul algebra, which portrays combinatorial correlation, in the end.


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## Body

## 1. Introduction and Definitions

Over the last few years, scholars have become increasingly reliant on computers for the majority of their research work, and the second fashion is graph theory, an extremely popular bough of mathematics. By interacting with several graphs, people can grasp numerous practical applications. In essence, the problem of Konigsberg bridge, happened in 1735, was the genesis of graph theory, and the researchers later worked significantly and intensively on the complete graph, bipartite graph, and Eulerian graph. Cauchy and L’Huilier were influential in introducing a strong area of mathematics, topology, succeeding Leonhard Euler’s work. In the field of theoretical chemistry, Arthur Cayley was the very first scientist to analyze trees to predict chemical composition. Sylvester initially introduced the word graph, a mathematical structure that can be used to model the relationship between objects, in his work, and Frank Harary published a heroic book on graph theory in 1969 to unify mathematicians, chemists, engineers, biologists, social scientists, and computer scientists. Under the shadow of basic graphs, now, we can understand yeast two-hybrid problem [1], microarrays and RNA-seq [2–4], major problems of discrete mathematics, and protein-protein interaction problem [5–9]. When dealing with chemical reaction networks (CRNs) [10], graph theory is a valuable, prolific, versatile, and companionable tool. It has unquestionably become an essential academic discipline in a variety of domains, including computation flow, GPS (route, track, and waypoint), communication networks, computer science, Google Maps, computational devices, Simulink [11, 12], and so on. It appears hard in today’s reality to describe characteristics of classical random graphs in relevance to representations of meaningful complex networks so bipartite graphs can be leveraged to solve this complicated problem [13] and assisting in the advance coding theory, database management, document/word problem, optimal assignment problem, communication network addressing, radar system, query log analysis, missile guidance, astronomy, personnel assignment problem, circuit design, crystallography, projective geometry [14], and x-ray [15, 16] (see Figure 1).Figure 1
A computer network with diagnostic links.The relationship in the form of turbo codes and low-density parity-check probabilistic decoding between a factor graph, a particular graph, and belief network are extremely close (see [17]). The captivating part of mathematics, chemical graph theory, by interrelating chemistry, is a new direction of modern research. In the form of a molecular graph, the molecules from chemistry are modelled mathematically. Vertices represent atoms in a molecular graph, whereas edges represent chemical bonds. These molecular structures are subjected to a variety of graph theory approaches in order to determine their topological and structural properties. The boiling point of a chemical compound, which is a physical entity, can be approximated using the degree and distance between the chemical compound’s vertices, for example. Thus, topology of the molecular structure plays a crucial role in predicting the compelling advantages of the accompanying chemical compound through mathematics [18]. In 1988, it was reported that a few hundred specialist analysts worked on delivering roughly 500 research publications each year, looking at various aspects of chemical structures, including Gutman’s two-volume fastidious material [19]. In [20–29], you can find more applications of this intriguing field of research, including discussions of topological indices. The study of chemical compounds in terms of mathematical modelling is one of the most recent research directions among scientists [27, 28].Chemical compounds with distinctive mathematical structures and a diverse variety of uses in industrial, medicinal, research, and commercial chemistry occur in large numbers. A chemical compound’s atom arrangements follow specific structural laws that have beneficial anomalous behavior. Thus, in applied research, adopting mathematical methods such as combinatorics and topology to investigate these attributes play a key role. It is fair to say that the discipline of chemical graph theory makes valuable strides to mathematical chemistry [20, 21]. Many chemical graph theory invariants, such as indices or descriptors, are applied in other sciences, particularly in the pharmaceutical and chemical industries [24, 25]. The research of distance-based and degree-based indices, in particular, plays an important role in the development of related subjects [30]. It aids in the collection of large amounts of data in the form of numerical values associated with chemical structures and the comparison of those values utilizing modern computer systems [31]. Many topological descriptors were introduced in the latter decade of the nineteenth century to meet the needs of chemists [32, 33].Mathematicians have made substantial progress in the study of Koszul algebras and their representations in the recent few decades. In commutative algebra, topology, algebraic geometry, and representation theory, they exploited it extensively. In the form of linear minimal graded free resolution, Jean-Louis Koszul, a French mathematician, introduced Koszul algebra. This resolution yields graded Betti numbers, which can be used to really need homological invariants of a module. Stewart explored Koszul resolutions for enormous classes of algebras, Steenrod algebra, and universal enveloping algebra in depth in 1970 (see [34]). Conca introduced Koszul filtrations in [35] as a result of enthusiastic work on strongly Koszul algebras in [36]. The Koszulness of the ring [37] is determined by the quadratic Gröbner basis of an ideal of the residue class ring; however, there are some Koszul algebras whose defining ideals are not generated quadratically under some monomial ordering. A toric ideal, a particular form of binomial ideal, combines combinatorics, geometry, and algebra and has a variety of applications, including contingency tables, integer programming, triangulations of convex polytopes, and algebraic geometry. The quadratic binomials generate a toric ideal with a finite graph combinatorially, and this leads us to believe that a Koszul algebra is normal if squarefree quadratic monomials generate the toric ideal but not the other way around [38]. Numerous classical algebra questions can be answered by connecting the three well-known boughs of mathematics, commutative algebra, graph theory, and linear algebra. The cyclotomic polynomial is connected to the circulant matrix, which is a specific type of Toeplitz matrix named after Otto Toeplitz. These special matrices have applications in linear algebra and graph theory. With the help of circulant matrices, the system of linear equations can be converted into circular convolution, and by using the circular convolution theorem, we can use the discrete Fourier transform to transform the cyclic convolution into componentwise multiplication. Researchers can call a graph is a circulant if the adjacency matrix of a simple finite graph is circulant. In other words, we can state that a graph is a circulant if its group of automorphisms comprises a full-length cycle. Möbius ladders, for example, are circulant graphs.A graphΓ=Σ,ϒ is made up of two sets, edge set ϒ and vertex set Σ. If both these sets have a finite number of items, then Γ is the finite graph. Otherwise, Γ is a graph that is infinite. A simple graph is one that has no loops and multiple edges and is undirected if the edges do not reflect directions. If there is an edge between two vertices x1 and x2 in the set Σ of the graph Γ, they are considered neighbours, and the edge is said to be an incident in this situation. A simple graph is connected if there is a path between any two vertices. A cycle is a simple graph with n≥3 vertices in which the sequence of edges is built by connecting the end vertex of one edge to the starting vertex of the next edge, denoted by Cn.An infinite ring containing members of the typea0+a1x+a2x2+⋯+anxn in one variable x is referred to as a polynomial ring, denoted as Kx, in which all the coefficients ai are drawn from the field K of characteristic 0 and powers are nonnegative integers. Similarly, R=Kx1,…,xn can be used to define and indicate a polynomial ring in n variables over the field K. K⊂Zℜ. A graded ring R is a ring that can be written as a direct sum K under the condition R, where the additive subgroups R0⊕R1⊕R2⋯ and elements of Rχ1+χ2⊇Rχ1Rχ2∀χ1,χ2>0 are known as homogeneous components of Rχ of degree Rχ and homogeneous of degree R, respectively. If an ideal χ, then χ is called a graded ideal.A graded rings homomorphism is a ring homomorphismϕ:ℜ1→ℜ2 of two graded rings ℜ1 and ℜ2 with ϕℜ1χ⊆ℜ2χ for χ=0,1,2,…, for R2. It can be shown easily that ℜ, ℜ0-module, and ℜ0-algebra have the subring R0. is called ℜ-algebra, if both ℜ and K⊂Zℜ (inclusion) have the same identity. The residue class ring Kx1,…,xn/Iℜ is isomorphic to any standard graded K-algebra where Iℜ is graded ideal. For example, With ℜ=Kx1,…,xn and degxi=1 for K, we say the prototype of a standard graded K-algebra is the polynomial ring of n variables. (1) Let the set Sℜ contains all the chains i=1,…,n of prime ideals in R, mathematically, in set builder form, this set is equal to {pseq=p0⊂⋯⊂pm⊂ℜ|pi prime ideal}. (2) If i=1,2,…,n, then length pseq=m. (3) The numerical value dim ℜ=suplengthpseq|pseq∈Sℜ is Krull dimension of R.The chain0⊂x1⊂x1,x2⊂⋯⊂x1,x2,…,xn convinces us to say that the Krull dimension of ℜ=Kx1,…,xn is n. Let / be any ideal of R then this notion, Krull dimension, based on prime ideals is the same ℜ/I and I. ℜ/I; its ideal is the same. The largest number ℷ associated with the chain p0⊂p1⊂p2⊂⋯⊂pℷ=p of prime ideals is taken as the height of p, and this number can be denoted by ht p=ℷ. Let us suppose K, a base field; ℜ, a standard graded K-algebra; and N, a nonzero finitely generated graded ℜ-module of dimension d with the graded components Ni (finite dimensional K-vector spaces of N). Any member of Ni is said to be homogeneous of degree i, and any member of N is the unique finite sum of homogeneous members.A function of numerical valuesh:ℤ→ℤ, defined by i↦dimKNi, is known as the Hilbert function of N. From this function, we can derive the series Tex translation faileddimKNiti known as Hilbert series of N, there is a polynomial, Laurent-polynomial QNt in ℤt,t−1 with QN1>0 and hNt=QNt/1−td and d−1 degree polynomial PNx, Hilbert polynomial, in ℚx with dimKNi=PNi for all i>degQN−d. The pole order of hNt at t=1 is said to be Krull dimension d of N at t=1. The multiplicity eN of N is the positive number QN1, and eN/d−1! is the leading coefficient of PNx of N. The number degQNt−d, degree of the Hilbert series hNt, is called a-invariant. Let QNt=∑i=r1r2hiti.The coefficient vectorhr1,hr1+1,…,hr2 of QNt is said to be the h-vector of N. Let us consider an ideal I of the ring ℜ. The maximal length n of an N-sequence b1,…,bn∈I is said to be the I-depth of N with the condition N≠IN and can be written as depthI,N. If N=IN then I-depth of N is by convention ∞. A pair m,ℜ of unique maximal ideal and commutative ring is taken to be the local ring. In the case of the local ring ℜ, we can say depth of N, depthN=depthm,N, is simply m-depth of N. The following Auslander–Buchsbaum theorem shows the relation between depth and projective dimension. In the presence of ℜ, Noetherian local ring, and N, finitely generated ℜ-module of finite projective dimension, the statement of Auslander–Buchsbaum theorem is depthN+pdℜN=depthℜ where the projective dimension of N in ℜ is pdℜN. N is Cohen–Macaulay if N is trivial and in case of nontrivial it must have depthN = dim N.An exact sequence0←N←F0←F1←F2←⋯, defined by φi:Fi→Fi−1 and ϵ:F0→N, is called graded ℜ-resolution of N where all Fi are graded free ℜ-modules generated by finite sets, and if ℜ=Kx1,…,xn, this sequence Gmin (say) is known as minimal graded free ℜ-resolution if φiFi⊂mFi−1∀i, where m is the graded maximal ideal x1,…,xn in ℜ. The dimension of the final free module in Gmin is known as the type of Cohen–Macaulay ℜ-module N. A Cohen–Macaulay ℜ-module N with type 1 is called Gorenstein. Let us consider above-mentioned Gmin with Fa=⊕bℜ−bβa,bN. The uniquely determined numbers βa,bN or βa,b by N are said to be graded Betti numbers of N. The graded Betti numbers play a vital role to deduce other important homological invariants of N. The projective dimension of N is the number pdℜN=maxa:βa,b≠0,for someb, and reg N=maxb:βa,a+b≠0,for somea is said to be the regularity of N, and depthN=mina:βn−a,b≠0,for someb is known as the depth of N. We say Gmin is linear if regN=αN, where αN indicates least degree of a generator of N. N is said to be Koszul algebra if Gmin is linear.A nonvacuous setA=v1,…,vm of the monomials in ℜ=Kt1,…,tn is called monomial configuration of ℜ. Then the toric ring Kv1,…,vm is the subring of ℜ denoted by KA. The toric ideal, defining ideal of KA, IA of A is the kernel of epimorphism ϕ:ℜ′→KA defined by ϕxi=vi for i=1,…,m, where ℜ′ is the polynomial ring Kx1,…,xm. Every toric ideal IA is a prime ideal. A polynomial v1−v2 in ℜ is called a binomial where v1 and v2 are monomials of ℜ and binomially generated ideal is called binomial ideal. IA is the binomial ideal generated by v1−v2 with ϕv1=ϕv2. A primitive binomial is the binomial f=v1−v2∈IA if there ∄g=u1−u2∈IA with the inequality g≠f such that u1|v1 and u2|v2. If A1 and A2 are the sets of primitive binomials and irreducible binomials, respectively, in ℜ, then A1⊂A2. If we have initial ideal in<I, generated by the monomials in<v1,in<v2,…,in<vs, of I, nonzero ideal of Kx1,…,xn, then a finite subset of nonzero polynomials v1,v2,…,vs⊂I is called the Gröbner basis or standard basis of I under any ordering <. With respect to monomial ordering <, Gröbner basis of I always exist, and every finite superset G′ of G, Gröbner basis of I, is also Gröbner basis of I under <. Nonzero polynomials u1,…,us form Gröbner basis if in<ui=in<vi, where G=v1,v2,…,vs is a Gröbner basis of I. A standard basis u1u2,…,us is considered to be reduced if leading coefficients of in<ui=in<vi for G=v1,v2,…,vs are unity I I, and in case of ∀1≤i≤s, v1,v2,…,vs does not divide any supp in<vi.Uniquely determined reduced (standard) Gröbner basis always exists. A reduced standard basis of toric idealIA consists of primitive binomials. Let us consider a finite connected graph, Γ=Σ,ϒ without loops and multiple edges, on the vertex set Σ=t1,t2,…,tq, the edge set ϒ=e1,e2,…,en and Kt is the polynomial ring Kt1,t2,…,tq. Now we can attach the squarefree quadratic monomial titj∈Kt with an edge e=ti,tj∈ϒ where ti,tj are the members of Σ. The KΓ, edge ring of Γ, is considered to be the toric subring of Kt, which is generated by te1,te2,…,ten. The toric ideal, defining ideal of KΓ, IΓ is the kernel of epimorphism ϕ:ℜ→KΓ defined by ϕxi=tei for i=1,…,n, where ℜ is the polynomial ring Kx1,…,xn.


## 2. Main Results

LetA=a1,…,an⊂ℤ2n be the set of vectors such that each ai has at least one nonzero entry. Let Kx1,…,xn be the polynomial ring in n variables over the field K of characteristic 0. Consider the semigroup homomorphism φ:ℤ2n→ℤ2n defined by u=u1,u2,…,un↦u1a1+u2a2+⋯+unan. The image of φ is the semigroup ℤ2A=α1a1+α2a2+⋯+αnan:α1,α2,…,αn∈ℤ2. The map φ lifts to a surjective homomorphism ϕ:Kx1,…,xn→Kt1,…,tn. The toric ideal IA associated with A is the kernel of the epimorphism ϕ:Kx1,…,xn→Kt1,…,tn, given by ϕxi=tai=t1a1,it2a2,i,…,tnan,i, where ai=a1,i,…,an,i. The image of ϕ is called toric ring denoted by KA. Here is a well-known classic result.Lemma 1.
IfCn be the cycle of length n>2, then for ai∈ℤ2n, we have dimKKA=rankA, where dimKKA denotes the vector space dimension.Proof. Ifw1 and w2 are two nodes of Cn, a path of length m from w1 to w2 is a sequence of nodes w1=ti1,…,tim+1=w2 of Cn such that tij,tij+1 is any link in Cn for all j=1,2,…,m. We define the path ideal of Cn, denoted by ImCn to be the ideal of Kt1,…,tn generated by the monomials of the form ti1ti2,…,tim+1 where ti1,ti2,…,tim+1 is a path in Cn.To each pathpiCn,m=ti1,ti2,…,tim+1 of length m in Cn, we associate a vector aiCn,m∈ℤ2n such that(1)ai,jCn,m=1,iftij∈piCn,m,0,otherwise.Since there arepiCn,m=ti1,ti2,…,tim+1 paths of length m in Cn, we set aiCn,m∈Z2n, and thus, we obtain required result.Theorem 1.
LetCn be a cycle of length n and IACn,m be the toric ideal associated to the path ideal ImCn. Then we have the following:(a)
If gcdn,m+1=1, then IACn,m=0.(b)
If gcdn,m+1=d with d≠1, then IACn,m=fi where fi=xdx2d⋯xkd−xixi+d⋯xi+k−1d=∏θ=1kxθd−xi+θ−1d for i=1,…,d−1. So, here, we have d−1 generators, and each of the generator has degree k=n/d.Proof.
Obviously, there aren paths of length m in Cn. Thus, the matrix a1Cn,m,a2Cn,m,…,anCn,m is a circulant matrix of order n given by(2)A=κ0κn−1…κ2κ1κ1κ0…κ3κ2...............κn−2κn−3…κ0κn−1κn−1κn−2…κ1κ0,where(3)κi=1,if0≤i≤m;0,otherwise.
The rank of circulant matrixA is equal to n−d (see [39]), where d is the degree of gcd ft,tn−1, where ft=1+t+t2+⋯+tm is the associated polynomial to the matrix A. Since tm+1−1=t−1ft, it is enough to consider gcd tm+1−1,tn−1.
Now,tn−1=∏d1/nΦd1t, where Φd1t is the minimal polynomial of a primitive d1th root of unity called cyclotomic polynomial. It is an irreducible polynomial in ℤt with root a∈ℂ with ad1=1 but ai≠1 for i=1,…,d1−1.(a)
Ifm+1 and n are relatively prime, then tm+1−1 and tn−1 have only a=1 as a common root. Hence, rank of A is n, thus by previous result, IACn,m=0.(b)
If gcdn,m+1=d with d≠1, then gcd tm+1−1,tn−1=d so by the previous lemma dimKKA=n−d. Let A=fi and monomial ordering < is lex with x1>x2>⋯>xn. The image of every element of A is zero under ϕ so A⊆IACn,m is obvious. It therefore suffices to show that each polynomial in IACn,m is a K-linear combination of these binomials fi. Suppose g∈IACn,m cannot be written as a K-linear combination of binomials. We choose a polynomial g with this property such that the initial term in<g=∏θ=1kxi+θ−1d for i=1,…,d−1, is minimal with respect to the term order < among all the elements of IACn,m.
When expandingϕgx1,…,xn=gta1,ta2,…,tan=gt1t2,…,tm+1,t2t3,…,tm+2,…,tnt1,…,tm, we get zero because g∈IACn,m. In particular, the term ϕ∏θ=1kxi+θ−1d=tφuθ must cancel during this expansion. Hence, there is some other monomial ∏θ=1kxθd appearing in g such that φuθ=φv. Also, the polynomial g′=g+∏θ=1kxi+θ−1d−∏θ=1kxθd cannot be written as a K-linear combination of binomials in IACn,m because g cannot be written as a K-linear combination of binomials. Hence, in<g′<in<g. This is the contradiction to the fact that in<g=∏θ=1kxi+θ−1d is minimal with respect to the term order < among all the elements of IACn,m.Now our next notion is the primary decomposition of the leading ideal of the toric ideal. First, we see an example in this case and then give a proposition.Example 1.
Consider the polynomial ringKx1,…,x15 and m+1=6. Then the primary decomposition of LIAC15,5 is given by(4)LIAC15,5=x1x4x7x10x13,x2x5x8x11x14,LIAC15,5=x1,x2∩x2,x4∩x1,x5∩x10,x14∩x13,x14∩x4,x5∩x2,x7∩x5,x7∩x1,x8∩x4,x8∩x7,x8∩x2,x10∩x5,x10∩x8,x10∩x1,x11∩x4,x11∩x7,x11∩x10,x11∩x2,x13∩x5,x13∩x8,x13∩x11,x13∩x1,x14∩x4,x14∩x7,x14.where LI is the leading ideal of ideal I under the lex monomial ordering < with x1>x2…xn.Proposition 1.
By the same notations, the primary decomposition of the leading idealLIACn,m=∩λ=1kd−1pλ consists of the following characteristics:(1)
Total number of primary components iskd−1(2)
Number of variables (generators) in each component isd−1(3)
xd,x2d,…,xkd are the missing variables in this presentation(4)
This presentation is irredundant so it is unique(5)
Total number of existing variablesn−k(6)
Total number of generators in the primary decomposition isd−1kd−1(7)
Each variable is appearingd−1kd−1/n−k times in the primary decompositionIn the remaining portion of this chapter, we shall useN for the residue ring Kx1,…,xn/IACn,m.Lemma 2.
By the same notations, dimN=n−d+1.Proof.
Let monomial ordering< is lex with x1>x2>⋯>xn. By primary decomposition LIACn,m=∩λ=1kd−1pλ, we can easily verify that the cardinality of A=fi for i=1,…,d−1 is equal to the height of LIACn,m. That is, htLIACn,m=d−1. So Krull dimension of Kx1,…,xn/LIACn,m is n−d−1=n−d+1. Thus, by Corollary 5.3.14. of [40], we have the result.Lemma 3.
With the previous notations, the generators∏θ=1kxθd−xi+θ−1d for i=1,…,d−1 of IACn,m are the primitive binomials with lex monomial ordering and x1>x2>⋯>xn.Proof.
By Buchberger algorithm, it is easy to findS-polynomial Sfp,fq of any two generators fp=∏θ=1kxθd−xp+θ−1d and fq=∏θ=1kxθd−xq+θ−1d for p≠q, 1≤p,q≤d−1, can be written as upfp+uqfq, where up=−∏θ=1kxθd=−uq. In other words, this S-polynomial reduces to zero through the set of generators A=fi for i=1,…,d−1. Thus, A is the Gröbner basis of the toric ideal IACn,m with lex monomial ordering and x1>x2>⋯>xn. It is obvious to say that elements of A are the reduced Gröbner basis, so by proposition 10.1.2 of [41], we can say IACn,m is generated by primitive binomials.Proposition 2.
With the same notations, the minimal graded free resolution ofℜ=Kx1,…,xn-module N is (5)N←NC0d−10←NC1d−11−k←NC2d−12−2k←⋯←NCd−1d−1d−11−k,with the graded Betti numbers(6)βi,j=βi,ki=Cid−1,ifi=0,…,d−1;0,otherwise.


## 3. Numerical Data Obtained from Graded Betti Numbers

In this section, we shall deduce the most important homological invariants of the finitely generated module from the graded Betti numbers.LetN be a finitely generated graded ℜ=Kx1,…,xn-module with Betti numbers βa,b=βa,bN where(7)βa,b=Cad−1,ifa=0,…,d−1;0,otherwise.The numbers(8)pdℜN=maxa:βa,b≠0for someb=d−1,(9)regN=maxb:βa,a+b≠0for somea=d−1k−1,are projective dimension and regularity of N, respectively. It is obvious from (8) that pdℜN<n, so by the Auslander–Buchsbaum theorem(10)depthN+pdℜN=n,we can write(11)depthN=mina:βn−a,b≠0for someb=n−d+1.Due to Lemma2 and equation (11), the ℜ=Kx1,…,xn-module N is Cohen–Macaulay. By Auslander–Buchsbaum theorem, we can say this module as Cohen–Macaulay because its projective dimension and codimension are equal as dimKx1,…,xn−dimN=n−n−d+1=d−1. Since N is Cohen–Macaulay, and its type is 1 so it is Gorenstein.Proposition 3.
Letℜ=Kx1,…,xn be the standard graded K-algebra and N be a ℜ-module with graded Betti numbers βa,b. Then hNt=1−tkd−1/1−tn with 1−tkd−1=∑a−1a∑bβa,btb. After dividing 1−tkd−1 by the maximum possible power of 1−t, then this series will be hNt=∑α=0k−1tαd−1/1−tn−d+1.A positive integern−d+1 in this proposition is the Krull dimension of N as discussed in Lemma 2, and the number ∑α=0k−11αd−1=kd−1 is multiplicity of N denoted by eN. Let QNt=∑i=0k−1d−1hiti=∑α=0k−1tαd−1. The h-vector h=h0,h1,h2,…,hk−1d−1 contains, respectively, ones, counting numbers, triangular numbers, … when d=2,3,4,… of the Pascal triangle. Note that 0=0N. The 0-invariant 0N of N is degree of hNt. Mathematically, 0N=k−1d−1−n−d+1=−k=deg1−tkd−1−n is the upper bound of maxb:βa,b≠0for somei−n=kd−1−n.We can display the graded Betti numbers ofℜ=Kx1,…,xn-module N by the diagram called Betti diagram. In Betti diagram 2, graded Betti number βa,a+b is at the position a,b; any nonzero graded Betti number lies inside the bounded region; and corner points represent the extremal Betti numbers see (Figure 2).Figure 2
Betti diagram.


## 4. A Class of Koszul Algebras

Koszul algebras, Artin–Schelter regular algebras of lower global dimension three [42], play a vital role in the study of combinatorics, algebra, topology, and mathematical physics initially defined by Priddy [34]. Generally, it is very difficult to detect whether a given algebra is Koszul or not. One approach is to compute the first few matrices in the resolution and to find if they are linear. If the matrices are not linear, then this algebra is not Koszul. A more efficient way to prove that an algebra is Koszul is given by various kinds of filtration arguments. A family F of ideals called a Koszul filtration of N has the properties listed below:(1)
F contains only ideals I generated by linear forms(2)
F also contains (0), zero ideal, and m, maximal ideal(3)
I/J is a cyclic module whose annihilator belongs to F with 0≠I,J∈F, and J⊂IKoszul filtration and strongly Koszul algebra were discussed, in a good manner, in [35, 36], respectively. Any ideal of Koszul filtration has linear resolution [35] leads to Koszul algebra, but this is the one-way statement [43] and a residue class ring is this algebra if its ideal accepts the standard basis of degree 2 by [37]. However, the defining ideal of a Koszul algebra may not have a quadratic Gröbner basis with respect to any monomial order. Also, in [44], the writers give an example of a binomial link ideal whose residue class ring has a Koszul filtration, while the ideal has no quadratic Gröbner basis related with graphs. In [45], authors worked associated with four cycles for the Koszul filtration of edge ring. We denote a vector n,d,k to represent ℜ=Kx1,…,xn -module N=Kx1,…,xn/IACn,m where gcdn,m+1=d and k=n/d. Then the vector (2d,d,2) shows a class of Kosul algebras due to the fact that IACn,m is generated by the standard basis of degree 2.Now, we conclude this article by giving a concrete example. Let us considerR=Kx1,…,x15-module N=Kx1,…,x15/IAC15,5. The rank of a circulant matrix(12)A=100000000011111110000000001111111000000000111111100000000011111110000000001111111000000000011111100000000001111110000000000111111000000000011111100000000001111110000000000111111000000000011111100000000001111110000000000111111.is dimKKA=rankA=n−d=15−3=12 Toric idealIAC15,5 is generated by the reduced Gröbner basis f1=x3x6x9x12x15−x1x4x7x10x13 and f2=x3x6x9x12x15−x2x5x8x11x14 so the Krull dimension of N=Kx1,…,x15/IAC15,5 is n−d+1=13. Minimal graded free resolution of N=Kx1,…,x15/IAC15,5 is (13)N←NC020←NC12−4←NC22−8,with the graded Betti numbers β0,0=1, β1,5=2 and β2,10=1, f2=x3x6x9x12x15−x2x5x8x11x14, and N=Kx1,…,x15/IAC15,5 that can be represented by Table 1.Table 1
Graded Betti numbers ofN.
N012010010002000300040205000600070008001The projective dimension, regularity, and depth ofN are d−1=2, d−1k−1=8, and n−d+1=13, respectively. Auslander–Buchsbaum theorem is verified for these numbers; N=Kx1,…,x15/IAC15,5 is Cohen–Macaulay and Gorenstein. The coefficients of h-vector h=1,2,3,4,5,4,3,2,1 are the counting numbers can be obtained from the Hilbert series hNt=1−t52/1−t15=1+2t+3t2+4t3+5t4+4t5+3t6+2t7+t8/1−t13 of degree −k=−5, and kd−1=25 is the multiplicity of N=Kx1,…,x15/IAC15,5.


## 5. Conclusion

This paper comprises the appositeness among linear algebra, commutative algebra, and graph theory in the form of some homological invariants, Hilbert series, Krull dimension, graded Betti numbers, depth, and so on, and a class of Koszul algebra by taking different path lengths of a cycle. We can explore this class of Koszul algebra more broadly in the future by correlating any suitable type of simple graphs.



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*Source: 4450488-2022-03-24.xml*