Gee Choon Lau

• Ph.D.
• Universiti Teknologi MARA Johor Branch, Segamat Campus

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f\colon E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic n...

In this paper, we introduce two new concepts namely bijective product k-cordial labeling and bijective product square k-cordial labeling and show that some standard graphs admit bijective product k-cordial labeling, where k = 2, 3. Also, we establish that the path, cycle, flower, helm and gear graphs are bijective product square 3-cordial graphs.

A graph G is analytic odd mean if there exist an injective function f : V → {0, 1, 3, . . . , 2q − 1} with an induced edge labeling f ∗ : E → Z such that for each edge uv with f(u) < f(v), f ∗ (uv) = ⎧ ⎨ ⎩ l f(v) 2−(f(u)+1) 2 2 m if f (u) 6= 0; l f(v) 2 2 m if f (u) = 0. is injective. Clearly the values of f ∗ are odd. We say that f is an analytic...

An edge labeling of a graph G = (V,E) is said to be local antimagic if there is a bijection f : E → {1,..., |E|} such that for any pair of adjacent vertices x and y, f +(x) ≠ f +(y), where the induced vertex label is f +(x) = 𝜮 f(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χla(G), is th...

Let $G=(V, E)$ be a connected graph. A bijection $f: E\to \{1, \ldots, |E|\}$ is called a local antimagic labeling if for any two adjacent vertices $x$ and $y$, $f^+(x)\neq f^+(y)$, where $f^+(x)=\sum_{e\in E(x)}f(e)$ and $E(x)$ is the set of edges incident to $x$. Thus a local antimagic labeling induces a proper vertex coloring of $G$, where the v...

Let \(G = (V,E)\) be a connected simple graph of order \(p\) and size \(q\). A graph \(G\) is called local antimagic (total) if \(G\) admits a local antimagic (total) labeling. A bijection \(g : E \to \{1,2,\ldots,q\}\) is called a local antimagic labeling of $ if for any two adjacent vertices \(u\) and \(v\), we have \(g^+(u) \ne g^+(v)\), where \...

An edge labeling of a graph [Formula: see text] is said to be local antimagic if it is a bijection [Formula: see text] such that for any pair of adjacent vertices [Formula: see text] and [Formula: see text], [Formula: see text], where the induced vertex label of [Formula: see text] is [Formula: see text] ([Formula: see text] is the set of edges inc...

Consider a simple connected graph \(G = (V,E)\) of order p and size q. For a bijection \(f : E \to \{1,2,\ldots,q\}\), let \(f^+(u) = \sum_{e\in E(u)} f(e)\) where \(E(u)\) is the set of edges incident to u. We say f is a local antimagic labeling of G if for any two adjacent vertices u and v, we have \(f^+(u) \ne f^+(v)\). The minimum number of dis...

Let G=(V,E) be a connected simple graph of order p and size q. A graph G is called local antimagic if G admits a local antimagic labeling. A bijection f:E→{1,2,…,q} is called a local antimagic labeling of G if for any two adjacent vertices u and v, we have f+(u)≠f+(v), where f+(u)=∑e∈E(u)f(e), and E(u) is the set of edges incident to u. Thus, any l...

Let \(G = (V,E)\) be a simple graph of order \(p\) and size \(q\). A graph \(G\) is called local antimagic (total) if \(G\) admits a local antimagic (total) labeling. A bijection \(g : E \to \{1,2,\ldots,q\}\) is called a local antimagic labeling of \(G\) if for any two adjacent vertices \(u\) and \(v\), we have \(g^+(u) \neq g^+(v)\), where \(g^+(...

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number...

Let $G = (V,E)$ be a connected simple graph. A bijection $f: E \rightarrow \{1,2,\ldots,|E|\}$ is called a local antimagic labeling of $G$ if $f^+(u) \neq f^+(v)$ holds for any two adjacent vertices $u$ and $v$, where $f^+(u) = \sum_{e\in E(u)} f(e)$ and $E(u$) is the set of edges incident to $u$. A graph $G$ is called local antimagic if $G$ admits...

We introduce a new concept in graph coloring motivated by the popular Sudoku puzzle. Let $G=(V,E)$ be a graph of order $n$ with chromatic number $\chi(G)=k$ and let $S\subseteq V.$ Let $\mathscr C_0$ be a $k$-coloring of the induced subgraph $G[S].$ The coloring $\mathscr C_0$ is called an extendable coloring if $\mathscr C_0$ can be extended to a...

An edge labeling of a connected graph [Formula: see text] is said to be local antimagic if it is a bijection [Formula: see text] such that for any pair of adjacent vertices [Formula: see text] and [Formula: see text], [Formula: see text], where the induced vertex label [Formula: see text], with [Formula: see text] ranging over all the edges inciden...

For a graph $ G $, we define a total $ k $-labeling $ \varphi $ is a combination of an edge labeling $ \varphi_e(x)\to\{1, 2, \ldots, k_e\} $ and a vertex labeling $ \varphi_v(x) \to \{0, 2, \ldots, 2k_v\} $, such that $ \varphi(x) = \varphi_v(x) $ if $ x\in V(G) $ and $ \varphi(x) = \varphi_e(x) $ if $ x\in E(G) $, then $ k = \, \mbox{max}\, \{k_e...

Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic if $G$ admits a local antimagic labeling. A bijection $f : E \to \{1,2,\ldots,q\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $f^+(u) \ne f^+(v)$, where $f^+(u) = \sum_{e\in E(u)} f(e)$,...

In this paper, we provide a correct proof for the lower bounds of the local antimagic chromatic number of the corona product of friendship and fan graphs with null graph respectively as in [On local antimagic vertex coloring of corona products related to friendship and fan graph, {\it Indon. J. Combin.}, 5(2) (2021) 110--121]. Consequently, we obta...

Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic if $G$ admits a local antimagic labeling. A bijection $f : E \to \{1,2,\ldots,q\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $f^+(u) \ne f^+(v)$, where $f^+(u) = \sum_{e\in E(u)} f(e)$,...

Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic (total) if $G$ admits a local antimagic (total) labeling. A bijection $g : E \to \{1,2,\ldots,q\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $g^+(u) \ne g^+(v)$, where $g^+(u) = \sum_{e...

An edge labeling of a connected graph G = (V, E) is said to be local antimagic if it is a Bijection f : E → {1, … ,|E|} such that for any pair of adjacent vertices x and y, f⁺(x) ≠ f⁺(y), where the induced vertex label f⁺(x) = Σf(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χla(G), is th...

For a graph G, we define a total k-labeling ϕ as a combination of an edge labeling ϕe(x) → {1, 2,. .. , ke} and a vertex labeling ϕv(x) → {0, 2,. .. , 2kv}, such that ϕ(x) = ϕv(x) if x ∈ V (G) and ϕ(x) = ϕe(x) if x ∈ E(G), where k = max {ke, 2kv}. The total k-labeling ϕ is called an edge irregular reflexive k-labeling of G, if for every two edges x...

An edge labeling of a graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label of $x$ is $f^+(x)= \sum_{e\in E(x)} f(e)$ ($E(x)$ is the set of edges incident to $x$). The local antimagic chromatic num...

In this paper, a global optimization algorithm namely Kerk and Rohanin’s Trusted Region is used to find the global minimizers by employing an interval technique; with it, the algorithm can find the region where a minimizer is located and will not get trapped in a local one. It is able to find the convex part within the non-convex feasible region. T...

An edge labeling of a connected graph G = (V, E) is said to be local antimagic if it is a bijection f : E → {1, … ,|E|} such that for any pair of adjacent vertices x and y, f⁺(x) ≠ f⁺(y), where the induced vertex label f⁺(x) = ∑f(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χla(G), is th...

Let [Formula: see text] be a graph of order [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a dominating set of [Formula: see text] if every vertex in [Formula: see text] is adjacent to at least one vertex of [Formula: see text]. The domination polynomial of [Formula: see text] is the polynomial [Formula: see text], wher...

Let G=(V(G),E(G)) be a simple, finite and undirected graph of order n. Given a bijection f:V(G)→{1,…,n}, and every edge uv in E(G), let S=f(u)+f(v) and D=|f(u)−f(v)|. The labeling f induces an edge labeling f′:E(G)→{0,1} such that for an edge uv in E(G), f′(uv)=1 if gcd(S,D)=1, and f′(uv)=0 otherwise. Such a labeling is called an SD-prime labeling...

An edge labeling of a connected (p, q) -graph G = (V, E) of order p and size q is a modulo local antimagic labeling if it is a bijection π : E → {1, … ,q} such that for any pair of adjacent vertices u and v, π⁺(u) ≠ π⁺(v), where the induced vertex label π⁺(u) = ∑π(e)(mod p), with e ranging over all the edges incident to u. The modulo local antimagi...

For any graph G of order p, a bijection f:V(G)→{1,2,…,p} is called a numbering of G. The strength strf(G) of a numbering f of G is defined by strf(G)=max{f(u)+f(v)|uv∈E(G)}, and the strength str(G) of a graph G is str(G)=min{strf(G)|fisanumberingofG}. In this paper, many open problems are solved, and the strengths of new families of graphs are dete...

For any graph $G$ of order $p$, a bijection $f: V(G)\to [1,p]$ is called a numbering of the graph $G$ of order $p$. The strength $str_f(G)$ of a numbering $f: V(G)\to [1,p]$ of $G$ is defined by $str_f(G) = \max\{f(u)+f(v)\; |\; uv\in E(G)\},$ and the strength $str(G)$ of a graph $G$ itself is $str(G) = \min\{str_f(G)\;|\; f \mbox{ is a numbering o...

Let G be a graph of order n. A subset S of V (G) is a dominating set of G if every vertex in V (G)\S is adjacent to at least one vertex of S. The domination polynomial of G is the polynomial D(G, x) = ∑ n i=γ(G) d(G, i)x i , where d(G, i) is the number of dominating sets of G of size i, and γ(G) is the size of a smallest dominating set of G, called...

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number...

An edge labeling of a connected graph \(G = (V, E)\) is said to be local antimagic if it is a bijection \(f:E \rightarrow \{1,\ldots ,|E|\}\) such that for any pair of adjacent vertices x and y, \(f^+(x)\not = f^+(y)\), where the induced vertex label \(f^+(x)= \sum f(e)\), with e ranging over all the edges incident to x. The local antimagic chromat...

An edge labeling of a connected graph $G = (V,E)$ is said to be local antimagic if it is a bijection $f : E \to \{1, . . . , |E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x) \ne f^+(y)$, where the induced vertex label $f^+(x) = \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic numb...

In this note, we show the existence of integer sequences of lengths at least 3 (except 7) such that for every integer in position $i\equiv 1\pmod{4}$ (respectively position $j\equiv 3\pmod{4}$), counting from left to right, the sum of the integer and the adjacent integer(s) has a constant sum $x$ (respectively $y$) with $x\ne y$.

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number...

An antimagic labelling of a graph $G$ is a bijection $h : E(G) \to \{1, \ldots, |E(G)|\}$ such that the induced vertex label $h^+(v) = \sum_{uv\in E(G)}$ distinguish all vertices $v$. A well-known conjecture of Hartsfield and Ringel (1994) is that every connected graph other than $K_2$ admits an antimagic labelling. In 2017, two sets of authors (Ar...

Let G be a graph with vertex set V (G) and edge set E(G). A (p, q)-graph G = (V, E) is said to be AL(k)-traversal if there exists a sequence of vertices (v1, v2,. .. , vp) such that for each i = 1, 2,. .. , p − 1, the distance between vi and vi+1 is k. We call a graph G a k-step Hamiltonian graph (or say it admits a k-step Hamiltonian cycle) if it...

Let G be a graph with vertex set V (G) and edge set E(G). A (p, q)-graph G = (V, E) is said to be AL(k)-traversal if there exists a sequence of vertices (v1, v2,. .. , vp) such that for each i = 1, 2,. .. , p − 1, the distance between vi and vi+1 is k. We call a graph G a k-step Hamiltonian graph (or say it admits a k-step Hamiltonian cycle) if it...

Let G = (V,E) be a graph with p vertices and q edges. A graph G is analytic odd mean if there exist an injective function f : V →{0,1,3,5...,2q − 1} with an induce edge labeling f ∗ : E → Z such that for each edge uv with f(u) < f(v),f∗ (uv) =lf(v) 2 −(f(u)+1) 22m, if f(u)6= 0lf(v) 22m, if f(u) = 0is injective. We say that f is an analytic odd m...

Let G = (V, E) be a graph with p vertices and q edges. A graph G is analytic odd mean if there exist an injective function

Let G = (V, E) be a graph with p vertices and q edges. A graph G is analytic odd mean if there exist an injective function f : V → {0, 1, 3, 5 . . . , 2q − 1} with an induce edge labeling f ∗ : E → Z such that for each edge uv with f (u) < f (v), f ∗(uv) = � f (v)2−( f (u)+1)2 2 � , i f f (u) , 0 � f (v)2 2 � , i f f (u) = 0 is injective....

Let G = (V,E) be a graph with p vertices and q edges. A graph G is an analytic odd mean if there exist an injective function f∶ V → {0,1,3,5 ...,2q - 1} with an induce edge labeling f^*: E → Z such that for each edge uv with f(u) < f(v), f^* (uv)={■((〖f(v)〗^2-〖(f(u)+1)〗^2)/2&if f(u)≠0@〖f(v)〗^2/2&if f(u)=0)} is injective. We say that f is an analyti...

Let P(G,k) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted G � H, if P(G,k) = P(H,k). We write [G] = {HOEH � G}. If [G] = {G}, then G is said to be chromatically unique. In this paper, we first characterize certain complete 5-partite graphs G with 5n vertices according to the number of 6...

Let P(G, �) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted by G ∼ H, if P(G, �) = P(H, �). We write [G] = {H|H ∼ G}. If [G] = {G}, then G is said to be chromatically unique. In this paper, we first characterize certain complete 5-partite graphs with 5n + 1 vertices according to the numb...

Let P(G, �) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted G ∼ H, if P(G, �) = P(H, �). We write [G] = {H|H ∼ G}. If [G] = {G}, then G is said to be chromatically unique. In this paper, we first characterize certain complete 5-partite graphs G with 5n + i vertices for i = 1, 2, 3 accord...

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $p$ and size $q$. For $k\ge 1$, a bijection $f: V(G)\cup E(G) \to \{k, k+1, k+2, \ldots, k+p+q-1\}$ such that $f(uv)= |f(u) - f(v)|$ for every edge $uv\in E(G)$ is said to be a $k$-super graceful labeling of $G$. We say $G$ is $k$-super graceful if it admits a $k$-super graceful...

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $p$ and size $q$. For $k\ge 1$, a bijection $f: V(G)\cup E(G) \to \{k, k+1, k+2, \ldots, k+p+q-1\}$ such that $f(uv)= |f(u) - f(v)|$ for every edge $uv\in E(G)$ is said to be a $k$-super graceful labeling of $G$. We say $G$ is $k$-super graceful if it admits a $k$-super graceful...

A (p, q) −graph G with vertex set V(G) and edge set E(G) is said to be AL(k) −traversable for k ≥ 1 if we can arrange its vertex set as the sequence of vertices {v1,v2, …, vp} such that the distance between vi and vi+1 for each i = 1,2, …, p −1 is k. A graph G is called k −step Hamiltonian if it is AL(k) −traversable and d(v1,vp) = k. Then, the seq...

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number...

A $(p,q,r)$-board that has $pq+pr+qr$ squares consists of a $(p,q)$-, a $(p,r)$-, and a $(q,r)$-rectangle such that every two of them share a common side of same length. Let $S$ be the set of the squares. Consider a bijection $f : S \to [1,pq+pr+qr]$. Firstly, for $1 \le i \le p$, let $x_i$ be the sum of all the $q+r$ integers in the $i$-th row of...

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number o...

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number...

Let G be a simple graph with vertex set V(G) and edge set E(G). Let ⟨ℤ2, +,*⟩ be a field with two elements. A vertex labeling f : V(G) → ℤ2 induces two edge labelings f⁺: E(G) → ℤ2 such that f⁺ (xy) = f(x) + f(y), whereas f* : E(G) → ℤ2 such that f* (xy) = f(x) f(y), for each edge xy ∈ E(G). For i ϵ ℤ2, let and . A labeling f of a graph G is said t...

An edge labeling of a connected graph G = (V,E) is said to be local antimagic if it is a bijection f : E → {1, . . . , |E|} such that for any pair of adjacent vertices x and y, f^+(x) \ne f^+(y), where the induced vertex label f^+(x) = \sum f(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by...

Let G=(V,E) be a simple, finite and undirected (p,q)-graph with p vertices and q edges. A graph G is Skolem odd difference mean if there exists an injection f:V(G)→(0,1,2,...,p+3q-3) and an induced bijection f*:E(G)→(1,3,5,...,2q-1) such that each edge uv (with f(u)>f(v)) is labeled with f*(uv)=f(u)-f(v)2. We say G is Skolem even difference mean if...

Let G= (V,E) be a (p,q)-graph. A bijection f: E → to{1,2,3,ldots,q} is called an edge-prime labeling if for each edge uv in E, we have GCD(f⁺(u),f⁺(v))=1 where f⁺(u) =Σuw∈Ef(uw). Moreover, a bijection f: E → {1,2,3,ldots,q} is called a semi-edge-prime labeling if for each edge uv in E, we have GCD(f⁺(u),f⁺(v))=1 or f⁺(u)=f⁺(v). A graph that admits...

For a graph G, let P(G; λ) denote the chromatic polynomial of G. Two graphs G and H are chromatically equivalent if they share the same chromatic polynomial. A graph G is chromatically unique if any graph chromatically equivalent to G is isomorphic to G. A K4- homeomorph is a subdivision of the complete graph K4. In this paper, we determine a famil...

Let G be a graph with vertex set V(G) and edge set E(G). A vertex labeling f:V(G)→Z2 induces an edge labeling f+:E(G)→Z2 defined by f+(xy)=f(x)+f(y), for each edge xy∈E(G). For i∈Z2, let vf(i)=|{v∈V(G):f(v)=i}| and ef(i)=|{e∈E(G):f+(e)=i}|. We say f is friendly if |vf(0)−vf(1)|≤1. We say G is cordial if |ef(1)−ef(0)|≤1 for a friendly labeling f. Th...

Let be a simple, finite and undirected graph of order and size . A bijection such that for every edge is said to be a -super graceful labeling of . We say is -super graceful if it admits a -super graceful labeling. For , the function is called a super graceful labeling and a graph is super graceful if it admits a super graceful labeling. In this pa...

Let G be a graph with vertex set V(G) and edge set E(G), a vertex labeling \(f : V(G)\rightarrow \mathbb {Z}_2\) induces an edge labeling \( f^{+} : E(G)\rightarrow \mathbb {Z}_2\) defined by \(f^{+}(xy) = f(x) + f(y)\), for each edge \( xy\in E(G)\). For each \(i \in \mathbb {Z}_2\), let \( v_{f}(i)=|\{u \in V(G) : f(u) = i\}|\) and \(e_{f^+}(i)=|...

For a graph $G$, let $P(G,\lambda)$ denote the chromatic polynomial of $G$. Two graphs $G$ and $H$ are chromatically equivalent if they share the same chromatic polynomial. A graph $G$ is chromatically unique if for any graph chromatically equivalent to $G$ is isomorphic to $G$. In this paper, the chromatically unique of a new family of 6-bridge gr...

Let G = (V (G),E(G)) be a simple, finite and undirected graph of order n. Given a bijection f : V (G) → {1, . . . , n}, we associate 2 integers S = f(u) + f(v) and D = |f(u) - f(v)| with every edge uv in E(G). The labeling f induces an edge labeling f' : E(G) → {0, 1} such that for any edge uv in E(G), f'(uv) = 1 if gcd(S,D) = 1, and f'(uv) = 0 oth...

For a graph G, let P(G,λ) denote the chromatic polynomial of G. Two graphs G and H are chromatically equivalent (or simply χ-equivalent), denoted by G ~ H, if P(G,λ) = P(H,λ). A graph G is chromatically unique (or simply χ-unique) if for any graph H such as H ~ G, we have H ≅ G, i.e. H is isomorphic to G. A K4-homeomorph is a subdivision of the com...

Let G be a graph of order n. A subset S of V(G) is a dominating set of G if every vertex in V(G)(minus 45 degree rule)S is adjacent to at least one vertex of S. The domination polynomial of G is the polynomial D(G,x)=∑i=γ(G)nd(G,i)xi, where d(G,i) is the number of dominating sets of G of size i, and γ(G) is the size of a smallest dominating set of...

For a graph G, let P(G; λ) denote the chromatic polynomial of G. Two graphs G and H are chromatically equivalent (or simply χequivalent), denoted by G ~ H, if P(G; λ) = P(H; λ). A graph G is chromatically unique (or simply χ unique) if for any graph H such as H ~ G, we have H ≅ G, i.e, H is isomorphic to G. A K4-homeomorph is a subdivision of the c...

An injective map f : E(G) → {±1,±2,…,±q} is said to be an edge pair sum labeling if the induced vertex labeling function f^*:V(G)→Z-{0} is defined by f^* (v)=∑_(e∊E_v)▒f(e) is one–to–one where Ev denotes the set of edges in G that are incident to v and f^* (V(G) ) is of the form {±k_1,±k_2,…,±k_(p/2) } for even p and {±k_1,±k_2,…,〖±k〗_((p-1)/2) }⋃{...

Let G be a graph with vertex set V and edge set E such that |V| = p and |E| = q. We denote this graph by (p, q)-graph. For integers k≥. 0, define a one-to-one map f from E to {k, k+ 1, . . . , k+ q - 1} and define the vertex sum for a vertex v as the sum of the labels of the edges incident to v. If such an edge labeling induces a vertex labeling in...

Abstract Let G = (V (G), E(G)) be a simple, finite and undirected graph of order n. Given a bijection f : V (G) ∪ E(G) → Zk such that for each edge uv ∈ E(G), f(u) + f(v) + f(uv) is constant (mod k).Let nf (i) be the number vertices and edges labeled by i under f. If |nf (i) − nf (j)| ≤ 1 for all 0 ≤ i < j ≤ k − 1, we say f is a k-totally magic cord...

For a graph G = (V(G), E(G)), an edge labeling function f: E(G) → {0, 1,…, k - 1}, where k is an integer, 2 ≤ k ≤\ E(G) |, induces a vertex labeling function f∗: V(G) → {0,1,…, k - 1} such that f∗(v) is the product of the labels of the edges incident to v (mod k). This function f is called a k-total edge product cordial (or simply k-TEPC) labeling...

We proved that P n + 1 m is total product cordial. We also give sufficient conditions for the graph to admit (or not admit) a product cordial labeling.

An injective map f : E(G) → {±1,±2, · · · ,±q} is said to be an edge pair sum labeling of a graph G(p, q) if the induced vertex function f* : V (G) → Z − {0} defined by f*(v) = (formula presented) is one-one, where Ev denotes the set of edges in G that are incident with a vetex v and f* (V (G)) is either of the form (formula presented) according as...

For a graph G, let P(G,λ) denote the chromatic polynomial of G. Two graphs G and H are chromatically equivalent (or simply χ-equivalent), denoted by G ∼ H, if P(G,λ) = P(H,λ). A graph G is chromatically unique (or simply χ-unique) if for any graph H such as H ∼ G, we have H ≅ G, i.e, H is isomorphic to G. A K 4-homeomorph is a subdivision of the co...

Let G = (V,E) be a simple connected graph. A vertex labeling of f: V -> {0,1}of G induces two edge labelings f(+),f* : E -> 1{0,1} defined by f(+)(xy) = f(x)+f(y) (mod 2) and f*(xy) = f(x)f(y) for each edge xy is an element of E. For i is an element of{0,1}, let v(f) (i) = vertical bar v is an element of V : f(v) = i}vertical bar, e(f)(+)(i) = vert...

For integer $k \geq 1$, a $(p, q)$-graph $G=(V, E)$ is said to admit an $A L(k)$-traversal if there exist a sequence of vertices $\left(v_1, v_2, \ldots, v_p\right)$ such that for each $i=1,2, \ldots, p-1$, the distance between $v_i$ and $v_{i+1}$ is $k$. We call a graph $k$-step Hamiltonian (or admits a $k$-step Hamiltonian tour) if it admits an $...

Let P(G; λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H; λ) = P(G; λ) implies H is isomorphic to G. In this paper, we determine the chromaticity of all Turán graphs with at most three edges deleted. As a by product, we found many families of chromatically unique graphs and chromatic equivalenc...

Let G be a graph with vertex set V(G) and edge set E(G), and let A be an abelian group. A labeling f: V (G) → A induces an edge labeling f*: E(G) → A defined by f*(xy) = f(x)+f(y), for each edge xy ∈ E(G). For i ∈ A, let vf (i) = |{v ∈ V(G): f(v) = i}| and ef (i) = |{e ∈ E(G): f*(e) = i}|. Let c(f) = {|ef (i)-ef (j)|: (i, j) ∈ A × A}. A labeling f...

Let P(G,λ) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted G ∼ H, if P(G,λ) = P(H,λ). We write [G] = {H∣H ∼ G}. If [G] = {G}, then G is said to be chromatically unique. In this paper, we first characterize certain complete 5-partite graphs G with 5n vertices according to the number of 6-...

Let P(G, Λ) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted G ∼ H, if P(G, Λ) = P(H, Λ). We write [G] = {H|H ∼ G}. If [G] = {G}, then G is said to be chromatically unique. In this paper, two new families of chromatically uniquecomplete 5-partite graphs G having 5n+4 vertices with certain...

Let P(G, λ) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted G ∼ H, if P(G, λ) = P(H, λ). We write [G] = {H|H ∼ G}. If [G] = {G}, then G is said to be chromatically unique. In this paper, two new families of chromatically unique complete 5-partite graphs G having 5n+4 vertices with certai...

The energy E(G) of a graph G is the sum of the absolute values of the eigenvalues of G. A dendrimer is an artificially manufactured or synthesized molecule built up from branched units called monomers. In this paper, using some inequalities, we obtain bounds for energy of certain polyphenylene dendrimers.

Let $G$ be a graph with vertex set V and edge set E such that |V| = p and |E| = q. For integers k\geq 0, define an edge labeling f : E \rightarrow \{k,k+1,....,k+q-1\} and define the vertex sum for a vertex $v$ as the sum of the labels of the edges incident to v. If such an edge labeling induces a vertex labeling in which every vertex has a constan...

Let G be a (p,g)-graph. Suppose an edge labeling of G given by f:E(G)→1,2,⋯,q is a bijective function. For a vertex V∈V(G), the induced vertex labeling of G is a function f*(V)=σf(uv) for all uv∈E(G). We say f*(V) the vertex sum of v. If, for all V∈V(G), the vertex sums equal to a constant Mod(k) where k≥2, then we say G admits a Mod(k)-edge-magic...

Let P(G, λ) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted by G ∼ H, if P(G, λ) = P(H, λ). We write [G] = {H|H ∼ G}. If [G] = {G}, then G is said to be chromatically unique. In this paper, we first characterize certain complete 5-partite graphs with 5n + 1 vertices according to the numb...

This article presents the deterministic system for susceptible-infective model for HIV. In this paper the homotopy analysis method is employed to compute an exact analytical approximation to the solution of the deterministic system for this model. We do a comparison between this method and Runge-Kutta method.

Let G be a graph with vertex set V(G) and edge set E(G) and let A be an abelian group. A labeling f: V(G)→ A induces an edge labeling f*: E(G)→ A defined by f*(xy)= f(x)+ f(y), for each edge xy∈ E(G). For i∈ A, let vf(i)= card{v∈ V(G): f(v)= i} and ef(i)=card{e∈ E(G): f*(e)= i}. Let c(f)= {| ef(i)- ef(j)|: (i,j)∈ AxA}. A labeling f of a graph G is...

Let P(G, λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H, λ) = P(G, λ) implies H is isomorphic to G. Liu et al. [Liu, R. Y., Zhao, H. X., Ye, C. F.: A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs. Discrete Math., 289, 175–179 (2004)], and Lau and Peng...

Let P(G,λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if, for any graph H, P(H,λ)=P(G,λ) implies H is isomorphic to G. In his Ph. D. thesis, H. X. Zhao (Theorems 5.4.2 and 5.4.3) proved that, for any positive integer t≥3, the complete t-partite graphs K(p-k,p,p,⋯,p) with p≥k+2≥4 and K(p-k,p-1,p,⋯,p) with p≥2k≥4 are...

For a graph G , let P ( G , λ ) be its chromatic polynomial. Two graphs G and H are chromatically equivalent, denoted G ∼ H , if P ( G , λ ) = P ( H , λ ) . A graph G is chromatically unique if P ( H , λ ) = P ( G , λ ) implies that H ≅ G . In this paper, we determine all chromatic equivalence classes of 2-connected ( n , n + 4 )-graphs with exactl...

Let P(G,λ) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted G∼H, if P(G,λ)=P(H,λ). We write [G]={H∣H∼G}. If [G]={G}, then G is said to be chromatically unique. In this paper, we first characterize certain complete 5-partite graphs G with 5n+i vertices for i = 1, 2, 3 according to the numb...

Let P(G, λ) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted G ∼ H, if P(G, λ) = P(H, λ). We write [G] = {H | H - G}. If [G] = {G}, then G is said to be chromatically unique. In this paper, we first characterize certain complete triparite graphs G according to the number of 4-independent...

Let $P(G,lambda)$ be a chromatic polynomial of a graph $G.$ Two graphs $G$ and $H$ aresaid to be chromatically equivalent, denoted $Gsim H,$ if $P(G,l)=P(H,l).$We write $[G]={H,|,Hsim G}.$ If $[G]={G},$ then $G$ is said to be chromatically unique.In this paper, we first characterize certain complete 4-partite graphs $G$ accordingly to the numberof...

Let P(G,λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H,λ)=P(G,λ) implies H is isomorphic to G. In this paper, we study the chromaticity of Turán graphs with deleted edges that induce a matching or a star. As a by-product, we obtain new families of chromatically unique graphs.

Let P(G,λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H,λ)=P(G,λ) implies H is isomorphic to G. For integers k≥0, t≥2, denote by K((t−1)×p,p+k) the complete t-partite graph that has t−1 partite sets of size p and one partite set of size p+k. Let K(s,t,p,k) be the set of graphs obtained from K((...

For a graph G, let P(G,λ) be its chromatic polynomial. Two graphs G and H are chromatically equivalent, denoted G∼H, if P(G,λ)=P(H,λ). A graph G is chromatically unique if P(H,λ)=P(G,λ) implies that H≅G. In this paper, we shall determine all chromatic equivalence classes of 2-connected (n,n+4)-graphs with three triangles and one induced 4-cycle, un...