Harris Kwong

• Ph.D.
• Professor (Full) at SUNY Fredonia

A simple graph G with p vertices is said to be vertex-Euclidean if there exists a bijection f : V ( G ) → { 1 , 2 , … , p } such that f ( v 1 ) + f ( v 2 ) > f ( v 3 ) for each C 3 -subgraph with vertex set { v 1 , v 2 , v 3 } , where f ( v 1 ) < f ( v 2 ) < f ( v 3 ) . More generally, the vertex-Euclidean deficiency of a graph G is the smallest in...

A simple graph G with p vertices is said to be vertex-Euclidean if there exists a bijection f : V ( G ) → { 1 , 2 , … , p } such that f ( v 1 ) + f ( v 2 ) > f ( v 3 ) for each C 3 -subgraph with vertex set { v 1 , v 2 , v 3 } , where f ( v 1 ) < f ( v 2 ) < f ( v 3 ) . More generally, the vertex-Euclidean deficiency of a graph G is the smallest in...

A simple graph G = (V, E) is said to be vertex Euclidean if there exists a bijection f from V to {1, 2,…,∣V∣} such that f(u) + f(v) > f(w) for each C3 subgraph with vertex set {u, v, w}, where f(u) < f(v) < f(w). The vertex Euclidean deficiency of a graph G, denoted μvEuclid(G), is the smallest positive integer n such that G ∪ Nn is vertex Euclidea...

In this series of papers, the primary goal is to enumerate Hamiltonian cycles (HC's) on the grid cylinder graphs $P_{m+1}\times C_n$, where $n$ is allowed to grow whilst $m$ is fixed. In Part~I, we studied the so-called non-contractible HC's. Here, in Part~II, we proceed further on to the contractible case. We propose two different novel characteri...

Here, in Part II, we proceeded further with the enumeration of Hamiltonian cycles (HC's) on the grid cylinder graphs of the form Pm+1?Cn, where n is allowed to grow and m is fixed. We proposed two novel characterisations of the contractible HC's. Finally, we made a conjecture concerning the dependency of the asymptotically dominant type of HC's on...

In a recent paper, we have studied the enumeration of Hamiltonian cycles (abbreviated HCs) on the grid cylinder graph P m+1 × C n , where m grows while n is fixed. In this sequel, we study a much harder problem of enumerating HCs on the same graph only this time letting n grow while m is fixed. We propose a characterization for non-contractible HCs...

In the studies that have been devoted to the protein folding problem, which is one of the great unsolved problems of science, some specific graphs, like the so-called triangular grid graphs, have been used as a simplified lattice model. Generation and enumeration of Hamiltonian paths and Hamiltonian circuits (compact conformations of a chain) are n...

Graph Theory International audience We study the enumeration of Hamiltonian cycles on the thin grid cylinder graph $C_m \times P_{n+1}$. We distinguish two types of Hamiltonian cycles, and denote their numbers $h_m^A(n)$ and $h_m^B(n)$. For fixed $m$, both of them satisfy linear homogeneous recurrence relations with constant coefficients, and we de...

We study the enumeration of Hamiltonian cycles on the thin grid cylinder graph Cm x Pn+1. We distinguish two types of Hamiltonian cycles depending on their contractibility (as Jordan curves) and denote their numbers hmnc (n) and hmc (n). For fixed m, both of them satisfy linear homogeneous recurrence relations with constant coefficients. We derive...

We study two sums involving the Stirling numbers and binomial coefficients. We find their closed forms, and discuss the connection between these sums.

An Alternate Proof of Sury’s Fibonacci–Lucas Relation

We study the recurrence relations and derive the generating functions of the entries along the rows and diagonals of the Catalan and Bell number difference tables.

We derive two formulas for the summation Σni=0Cr+iDr+i, where both C k and Dk satisfy the same generalized second-order recurrence. They lead to many summation and product formulas for Fibonacci-type, Pell-type, and Jacobsthal-type numbers.

Let G=(V,E) be a connected simple graph. A labeling f:V→Z 2 induces two edge labelings f + ,f * :E→Z 2 defined by f + (xy)=f(x)+f(y) and f * (xy)=f(x)f(y) for each xy∈E. For i∈Z 2 , let v f (i)=|f -1 (i)|, e f + (i)=|(f + ) -1 (i)| and e f * (i)=|(f * ) -1 (i)|. A labeling f is called friendly if |v f (1)-v f (0)|≤1. For a friendly labeling f of a...

The authors study the recurrence relations and derive the generating functions of the entries along the rows and diagonals of the Catalan and Bell number difference tables.

Let G = (V,E) be a simple graph. A vertex labeling f: V → {0, 1} induces a partial edge labeling f*: E → {0, 1} defined by f*(uv) = f(u) if and only if f(u) = f(v). For i = 0, 1, let v f (i) = {pipe}{v ∈ V: F(v) = i}{pipe}, and ef (i) = {pipe}{e ∈ E: f*(e) = i}{pipe}. A graph G is uniformly balanced if {pipe}e f (0) - e f (1){pipe} ≤ 1 for any vert...

Many integrals require two successive applications of integration by parts. During the process, another integral of similar type is often invoked. We propose a method which can integrate these two integrals simultaneously. All we need is to solve a linear system of equations.

Let G be a simple graph. Any vertex labeling f:V(G)→ℤ 2 induces an edge labeling f * :E(G)→ℤ 2 according to f * (xy)=f(x)+f(y). For each i∈ℤ 2 , define v f (i)=|{v∈V(G):f(v)=i}|, and e f (i)=|{e∈E(G):f * (e)=i}|. The friendly index set of the graph G is defined as {|e f (0)-e f (1)|:|v f (0)-v f (1)|≤1}. We determine the friendly index sets of conn...

A vertex labeling f:V→{0,1} of the simple graph G=(V,E) induces a partial edge labeling f * :E→{0,1} defined by f * (uv)=f(u) if and only if f(u)=f(v). Let v(i) and e(i) be the number of vertices and edges, respectively, that are labeled i, and define the balance index set of G as {|e(0)-e(1)|:|v(0)-v(1)|≤1}. We determine the balance index sets of...

A vertex labeling f: V → ℤ2 of a simple graph G = (V, E) induces two edge labelings f +, f*: E → ℤ2 defined by f +(uυ) = f(u) + f(υ) and f*(uυ) = f(u)f(υ). For each i ∈ ℤ2, let υ f (i) = |{υ ∈ V: f(υ) = i}|, e f +(i) = |{e ∈ E: f +(e) = i}| and e*f (i) = |{e ∈ E: f*(e) = i}|. We call f friendly if |υ f (0) − υ f (1)| ≤ 1. The friendly index set an...

Define the sequence {Un} as Uo = 0, U1 = 1, and Un = pUn-1 - Un-2 for n > 2. We study Σh=0n hm (nh) Uh and Σh=0n (-1) n+h hm(nh) Uh, and express them in terms of two associated sequences. Special cases of p = 2, 3 lead to interesting binomial and Fibonacci identities.

Let G=(V,E) be a simple graph, and let A={0,1}. Any edge labeling f:E→A induces a partial vertex labeling f * :V→A that assigns 0 or 1 to f * (v), depending on whether there are more 0- or 1-edges incident to v, and leaves f * (v) unlabeled otherwise. For each i∈A, let e f (i) and v f (i) denote the number of edges and vertices, respectively, that...

In a recent article, Hirschhorn found the generating functions of two sequences introduced by Fahr and Ringel. We use a matrix method to obtain the same results in a simpler and more direct manner.

Given positive integers m and n, let S m n be the m-colored multiset {1 m , 2 m , . . . , n m }, where i m denotes m copies of i, each with a distinct color. This paper discusses two types of combinatorial identities associated with the permutations and combinations of S m n . The first identity provides, for m ≥ 2, an (m − 1)-fold sum for mn n . T...

Let G(V,E) be a simple graph. A vertex labeling f:V→ℤ 2 induces an edge labeling f * :E→ℤ 2 defined by f * (xy)=f(x)f(y) for each edge xy∈E. For each i∈ℤ 2 , let v f (i)={v∈V:f(v)=i}, and e f (i)={e∈E:f * (e)=i}. A vertex labeling f of G that satisfies the condition |v f (0)-v f (1)|≤1 is said to be friendly. The product-cordial index set of the gr...

Any vertex labeling f:V→{0,1} of the graph G=(V,E) induces a partial edge labeling f * :E→{0,1} defined by f * (uv)=f(u) if and only if f(u)=f(v). The balance index set of G is defined as {|f *-1 (0)-f *-1 (1)|∣|f -1 (0)-f -1 (1)|≤1}. In this paper, we propose a new and easier approach to find the balance index set of a graph. This new method makes...

Let G be a graph with vertex set V and edge set E. A labeling f:V→{0,1} induces a partial edge labeling f * :E→{0,1} defined by f * (xy)=f(x) if and only if f(x)=f(y) for each edge xy∈E. The balance index set of G, denoted BI(G), is defined as {|f *-1 (0)-f *-1 (1)|:|f -1 (0)-f -1 (1)|≤1}. In this paper, we study the balance index sets of graphs wh...

Any vertex labeling fV→{0,1} of the graph G=(V,E) induces a partial edge labeling f * :E→{0,1} defined by f * (uv)=f(u) if and only if f(u)=f(v). The balance index set of G is defined as {|f *-1 (0)-f *-1 (1)||f -1 (0)-f -1 (1)|≤1}. In this paper, we first determine the balance index sets of rooted trees of height not exceeding two, thereby complet...

Any edge labeling f:E→{0,1} of a simple graph C=(V,E) induces a vertex labeling f * :V→{0,1} defined by f * (x)=i if x is incident to more i-edges than (1-i)-edges, and f * (x) is unlabeled if x is incident to an equal number of 0- and 1-edges. Denote by e f (i) and v f (i) the number of edges and vertices, respectively, labeled i. We call f edge-f...

Let G be a graph with vertex set V and edge set E, and let A be an abelian group. A labeling f : V --> A induces an edge labeling f* : E --> A defined by f*(xy) = f (x) + f (y). For i is an element of A, let v(f)(i) = card {v is an element of V : f (v) = i} and e(f)(i) = card {e is an element of E : f*(e) = i}. A labeling f is said to be A-friendly...

Let G = (V, E) be a graph, a vertex labeling f : V -> Z(2) induces an edge labeling f* : E -> Z(2) defined by f* (xy) = f (x) +f (y) for each xy is an element of E. For each i is an element of Z(2), define v(f)(i) = vertical bar f(-1)(i)vertical bar and e(f)(i) = vertical bar f *(-1) (i)vertical bar. We call f friendly if vertical bar v(f) (1) - v(...

Let G=(V,E) be a graph with a vertex labeling f:V→ℤ 2 that induces an edge labeling f * :E→ℤ 2 defined by f * (xy)=f(x)+f(y). For each i∈ℤ 2 , let v f (i)=card{v∈V:f(v)=i} and e f (i)=card{e∈E:f * (e)=i}. A labeling f of a graph G is said to be friendly if |v f (0)-v f (1)|≤1. The friendly index set of G is defined as {|e f (1)-e f (0)|: the vertex...

Let G be a graph with vertex set V(G) and edge set E(G), and let A={0,1}. A labeling f:V(G)→A induces an edge partial labeling f * :E(G)→A defined by f * (xy)=f(x) if and only if f(x)=f(y) for each edge xy∈E(G). For each i∈A, let v f (i)=|{v∈V(G):f(v)=i}| and e f (i)=|{e∈E(G):f * (e)=i}. The balance index set of G, denoted BI(G), is defined as {|e...

In this short note, we study two families of determinants the entries of which are linear functions of Fibonacci or Lucas numbers. The results are rather simple, and the two determinants only differ by a constant.

Let a, b be two positive integers. A (p, q)-graph G is said to be Q(a)P (b)-super edge-graceful, or simply (a, b)-SEG, if there exist onto mappings f: E(G) → Q(a) and f ∗ : V (G) → P (b), where j {±a, ±(a + 1),..., ±(a + (q − 2)/2)} if q is even, Q(a) = {0, ±a, ±(a + 1),..., ±(a + (q − 3)/2)} if q is odd, j {±b, ±(b + 1),..., ±(b + (p − 2)/2)} if p...

Let G be a graph with vertex set V and edge set E. Any vertex labeling f:V→{0,1} induces a partial edge labeling f * :E→{0,1} defined by f * (xy)=f(x) if and only if f(x)=f(y). Let v f (i)=|f -1 (i) and e f (i)=|f *-1 (i)|. We call f a friendly labeling if |v f (0)-v f (1)|≤1. The balance index set of G is defined as BI(G)={|e f (0)-e f (1|:f is fr...

We study properties of arithmetic progressions consisting of three squares; in particular, how one arithmetic progression generates infinitely many others, by means of explicit formulas as well as a matrix method. This suggests an equivalence relation could be defined on the arithmetic progressions, which lead to interesting problems for further st...

Please submit solutions and problem proposals to Dr. Harris Kwong, Department of Mathematical Sciences, SUNY Fredonia, Fredonia, NY, 14063, or by email at [email protected] If you wish to have receipt of your submission acknowledged by mail, please include a self-addressed, stamped envelope. Each problem or solution should be typed on separate shee...

Suppose G = (V,E) is a graph with vertex set V and edge set E. A vertex labeling f: V --> (0, 1) induces an edge labeling f*: E --> (0, 1) defined by f*(ny) = If(x)- f(y)l. For i is an element of (0, 1), let v(f)(i) and e(f)(i) be the number of vertices v and edges e with f(v) = i and f*(e) = i, respectively. A graph G is cordial if there exists a...

A matrix method is used to determine the number of Hamiltonian cycles Pm × Pn, m = 4, 5. This provides an alternative to other approaches which had been used to solve the problem. The method and its more generalized version, transfer-matrix method, may give easier solution to cases in which m ⩾ 6.

We derive bounds for f(v), the maximum number of edges in a graph on v vertices that contains neither three-cycles nor four-cycles. Also, we give the exact value of f(v) for all v up to 24 and constructive lower bounds for all v up to 200. © 1993 John Wiley & Sons, Inc.

We derive bounds for f (v), the maximum number of edges in a graph on v vertices that contains neither three-cycles nor four-cycles. Also, we give the exact value of f (v) for all v up to 24 and constructive lower bounds for all v up to 200.

Let f(v) denote the maximum number of edges in a graph of order v and of girth at least 5. In this paper, we discuss algorithms for constructing such extremal graphs. This gives constructive lower bounds of f(v) for v • 200. We also provide the exact values of f(v) for v • 24, and enumerate the extremal graphs for v • 10.

Given a simple graph G=(V,E), a subset S of V is called a ‘neighbourhood set’ provided G is the union of the subgraphs induced by the closed neighbourhoods of the vertices in S. The minimum and maximum cardinalities amon all minimal neighbourhood sets of G are denoted by n(G) and N(G), respectively; n(G) is called the ‘neighbourhood number’ of G. I...

We study properties of the periodicity of an infinite integer sequence (mod M) generated by , where f(x) ∈ Z[x] and f(0) = 1. In particular, we consider the case in which f(x) is a product of t polynomials, all of them congruent to φ(x)5 modulo p, where φ(x) is irreducible modulo p. Consequently, we determine the minimum periods, modulo pN, of the...