• Korean Journal of Mathematics
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• 제27권2호
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• pp.525-534
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• 2019

Let $P(G,{\lambda})$ denote the number of proper vertex colorings of G with ${\lambda}$ colors. The chromatic polynomial $P(C_n,{\lambda})$ for the cycle graph $C_n$ is well-known as $$P(C_n,{\lambda})=({\lambda}-1)^n+(-1)^n({\lambda}-1)$$ for all positive integers $n{\geq}1$. Also its inductive proof is widely well-known by the deletion-contraction recurrence. In this paper, we give this inductive proof again and three other proofs of this formula of the chromatic polynomial for the cycle graph $C_n$.

• 대한수학회보
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• 제33권1호
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• pp.75-86
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• 1996

Let G be a finite simple connected graph with vertex set V(G) and edge set E(G). Let A(G) be the adjacency matrix of G. The characteristic polynomial of G is the characteristic polynomial $\Phi(G;\lambda) = det(\lambda I - A(G))$ of A(G). A zero of $\Phi(G;\lambda)$ is called an eigenvalue of G.

• 대한수학회보
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• 제55권2호
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• pp.625-631
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• 2018

Let R be a commutative ring with identity and let R[x] be the collection of polynomials with coefficients in R. There are a lot of multiplications in R[x] such that together with the usual addition, R[x] becomes a ring that contains R as a subring. These multiplications are from a class of functions ${\lambda}$ from ${\mathbb{N}}_0$ to ${\mathbb{N}}$. The trivial case when ${\lambda}(i)=1$ for all i gives the usual polynomial ring. Among nontrivial cases, there is an important one, namely, the case when ${\lambda}(i)=i!$ for all i. For this case, it gives the well-known Hurwitz polynomial ring $R_H[x]$. In this paper, we completely determine the Krull dimension of $R_H[x]$ when R is a $Pr{\ddot{u}}fer$ domain. Let R be a $Pr{\ddot{u}}fer$ domain. We show that dim $R_H[x]={\dim}\;R+1$ if R has characteristic zero and dim $R_H[x]={\dim}\;R$ otherwise.

• Kyungpook Mathematical Journal
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• 제48권2호
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• pp.233-240
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• 2008

We present a quantum theorical background of the existence of multi-variable link invariants, for example the Kauffman polynomial, by observing the quantum (sl(2,$\mathbb{C}$), ad)-invariant from the Kontsevich invariant point of view. The background implies that the Kauffman polynomial can be studied by using the sl(N,$\mathbb{C}$)-skein theory similar to the Jones polynomial and the HOMFLY polynomial.

• Journal of applied mathematics & informatics
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• 제25권1_2호
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• pp.155-167
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• 2007

A graph G is called to be a 2-degree integral subgraph of a q-tree if it is obtained by deleting an edge e from an integral subgraph that is contained in exactly q - 1 triangles. An added-vertex q-tree G with n vertices is obtained by taking two vertices u, v (u, v are not adjacent) in a q-trees T with n - 1 vertices such that their intersection of neighborhoods of u, v forms a complete graph $K_{q}$, and adding a new vertex x, new edges xu, xv, $xv_{1},\;xv_{2},\;{\cdots},\;xv_{q-4}$, where $\{v_{1},\;v_{2},\;{\cdots},\;v_{q-4}\}\;{\subseteq}\;K_{q}$. In this paper we prove that a graph G with minimum degree not equal to q - 3 and chromatic polynomial $$P(G;{\lambda})\;=\;{\lambda}({\lambda}-1)\;{\cdots}\;({\lambda}-q+2)({\lambda}-q+1)^{3}({\lambda}-q)^{n-q-2}$$ with $n\;{\geq}\;q+2$ has and only has 2-degree integral subgraph of q-tree with n vertices and added-vertex q-tree with n vertices.

• 대한수학회지
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• 제34권4호
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• pp.871-894
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• 1997

For a lifted nontrivial additive character $\lambda'$ and a multiplicative character $\chi$ of the finite field with $q^2$ elements, the 'Gauss' sums $\Sigma\lambda'$(tr $\omega$) over $\omega$ $\in$ SU(2n + 1, $q^2$) and $\Sigma\chi$(det $\omega$)$\lambda'$(tr $\omega$) over $\omega$ $\in$ U(2n + 1, $q^2$) are considered. We show that the first sum is a polynomial in q with coefficients involving certain new exponential sums and that the second one is a polynomial in q with coefficients involving powers of the usual twisted Kloosterman sums and the average (over all multiplicative characters of order dividing q-1) of the usual Gauss sums. As a consequence we can determine certain 'generalized Kloosterman sum over nonsingular Hermitian matrices' which were previously determined by J. H. Hodges only in the case that one of the two arguments is zero.

• Kyungpook Mathematical Journal
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• 제56권3호
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• pp.657-668
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• 2016

Given a partition ${\lambda}=({\lambda}_1,{\lambda}_2,{\ldots},{\lambda}_l)$ of a positive integer n, let Tab(${\lambda}$, k) be the set of all tabloids of shape ${\lambda}$ whose weights range over the set of all k-compositions of n and ${\mathcal{OP}}^k_{\lambda}_{rev}$ the set of all ordered partitions into k blocks of the multiset $\{1^{{\lambda}_l}2^{{\lambda}_{l-1}}{\cdots}l^{{\lambda}_1}\}$. In [2], Butler introduced an inversion-like statistic on Tab(${\lambda}$, k) to show that the rank-selected $M{\ddot{o}}bius$ invariant arising from the subgroup lattice of a finite abelian p-group of type ${\lambda}$ has nonnegative coefficients as a polynomial in p. In this paper, we introduce an inversion-like statistic on the set of ordered partitions of a multiset and construct an inversion-preserving bijection between Tab(${\lambda}$, k) and ${\mathcal{OP}}^k_{\hat{\lambda}}$. When k = 2, we also introduce a major-like statistic on Tab(${\lambda}$, 2) and study its connection to the inversion statistic due to Butler.

• 대한수학회지
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• 제8권1호
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• pp.25-30
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• 1971

Put ($$^*$$) $$G[x,y]={\sum}\limits^{p+q=n}_{p,q=0}[-n]_{p+q}c_{p,q}x^py^q$$, where $[{\lambda}]_m$ is the Pocbhammer symbol and the $c_{p,q}$ are arbitrary constants. Making use of the specialized forms of some of his earlier results (see [8] and [9] the author derives here bilateral generating functions of the type ($$^{**}$$) $${\sum}\limits^{\infty}_{n=0}{\frac{[\lambda]_n}{n!}}_2F_1[\array{{\rho}-n,\;{\alpha};\\{\lambda}+{\rho};}x]\;G[y,z]t^n$$ where ${\alpha}$, ${\rho}$ and ${\lambda}$ are arbitrary complex numbers. In particular, it is shown that when G[y, z] is a double hypergeometric polynomial, the right-band member of ($^{**}$) belongs to a class of general triple hypergeometric functions introduced by the author [7]. An interesting special case of ($^{**}$) when ${\rho}=-m,\;m$ being a nonnegative integer, yields a class of bilateral generating functions for the Jacobi polynomials $\{P_n{^{{\alpha},{\beta}}}(x)\}$ in the form ($$^{***}$$) $${\sum\limits^{\infty}_{n=0}}\(\array{m+n\\n}\)P{^{({\alpha}-n,{\beta}-n)}_{m+n}(x)\;G[y,z]{\frac{t^n}{n!}}$$, which provides a unification of several known results. Further extensions of ($^{**}$) and ($^{***}$) with G[y, z] replaced by an analogous multiple sum $H\[y_1,{\cdots},y_m\]$ are also discussed.

• 대한수학회논문집
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• 제12권3호
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• pp.603-617
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• 1997

Consider a Sobolev inner product on the space of polynomials such as $$ \phi(p,q) = \lambda p(c)q(c) + <\tau,p'(x)q'(x)> $$ where $\tau$ is a moment functional and c and $\lambda$ are real constants. We investigate properties of orthogonal polynomials relative to $\phi(\cdot,\cdot)$ and give necessary and sufficient conditions under which such Sobolev orthogonal polynomials satisfy a spectral type differential equation with polynomial coefficients.

• Journal of applied mathematics & informatics
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• 제11권1_2호
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• pp.109-123
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• 2003

A tree is called starlike if it has exactly one vertex of degree greate. than two. In [4] it was proved that two starlike trees G and H are cospectral if and only if they are isomorphic. We prove here that there exist no two non-isomorphic Laplacian cospectral starlike trees. Further, let G be a simple graph of order n with vertex set V(G) : {1,2, …, n} and let H = {$H_1$, $H_2$, …, $H_{n}$} be a family of rooted graphs. According to [2], the rooted product G(H) is the graph obtained by identifying the root of $H_{i}$ with the i-th vertex of G. In particular, if H is the family of the paths $P_k_1,P_k_2,...P_k_2$ with the rooted vertices of degree one, in this paper the corresponding graph G(H) is called the sunlike graph and is denoted by G($k_1,k_2,...k_n$). For any $(x_1,x_2,...,x_n)\;\in\;{I_*}^n$, where $I_{*}$ = : {0,1}, let G$(x_1,x_2,...,x_n)$ be the subgraph of G which is obtained by deleting the vertices $i_1,i_2,...i_j\;\in\;V(G)\;(O\leq j\leq n)$, provided that $x_i_1=x_i_2=...=x_i_j=o.\;Let \;G[x_1,x_2,...x_n]$ be characteristic polynomial of G$(x_1,x_2,...,x_n)$, understanding that G[0,0,...,0] $\equiv$1. We prove that $G[k_1,k_2,...,k_n]-\sum_{x\in In}[{\prod_{\imath=1}}^n\;P_k_i+x_i-2(\lambda)](-1)...G[x_1,x_2,...,X_n]$ where x=($x_1,x_2,...,x_n$);G[$k_1,k_2,...,k_n$] and $P_n(\lambda)$ denote the characteristic polynomial of G($k_1,k_2,...,k_n$) and $P_n$, respectively. Besides, if G is a graph with $\lambda_1(G)\;\geq1$ we show that $\lambda_1(G)\;\leq\;\lambda_1(G(k_1,k_2,...,k_n))<\lambda_1(G)_{\lambda_1}^{-1}(G}$ for all positive integers $k_1,k_2,...,k_n$, where $\lambda_1$ denotes the largest eigenvalue.