Graph Cn Research Articles

Study on Necessary and Sufficient Conditions for Euler Graph and Hamilton Graph

Three theorems are proposed in this paper. The first theorem is that a connected undirected graph G is an Euler graph if and only if G can be expressed as the union of two circles without overlapped sides. Namely, equation satisfies. The second theorem is that a connected simple undirected graph is a Hamilton graph if and only if G contains a sub-graph generated by union of circles of sub-graphs derived from two endpoints of common side. Namely, the equation satisfies (meaning of symbols in the equations see main body of this paper). The third theorem is that a connected simple undirected graph is a Hamilton graph if and only if the loop sum of two circles, and, of sub-graphs derived from two endpoints of common side in graph G is a sub-graphs of loop graph Cn.

The Pfaffian property of circulant graphs

The importance of the Pfaffian property of a graph stems from the fact that if the graph is Pfaffian, then the number of its perfect matchings can be computed in polynomial time. A graph G is Pfaffian if there exists an orientation of G, denoted by G⃗, such that the determinant of the skew adjacency matrix of G⃗ equals the square of the number of perfect matchings of G. An undirected graph G=(V,E) with n vertices is a circulant graph, denoted by Cn(a1,a2,…,am), if there exists a labeling of the vertices of G, v1,v2,…,vn, and m integers, a1,a2,…,am, such that the edge set E={vivj:i−j≡±ak(modn)for 1≤k≤m}. In this paper, the Pfaffian property of circulant graphs is completely characterized, that is, a simple connected circulant graph Cn(a1,a2,…,am) of even order is Pfaffian if and only if m=1 or, m=2 and a1+a2 is odd.

Bounds on the Size of the Minimum Dominating Sets of Some Cylindrical Grid Graphs

Let γPm □ Cn denote the domination number of the cylindrical grid graph formed by the Cartesian product of the graphs Pm, the path of length m, m≥2, and the graph Cn, the cycle of length n, n≥3. In this paper we propose methods to find the domination numbers of graphs of the form Pm □ Cn with n≥3 and m=5 and propose tight bounds on domination numbers of the graphs P6 □ Cn, n≥3. Moreover, we provide rough bounds on domination numbers of the graphs Pm □ Cn, n≥3 and m≥7. We also point out how domination numbers and minimum dominating sets are useful for wireless sensor networks.

ALGEBRAIC CONNECTIVITY OF LOLLIPOP GRAPHS: A NEW APPROACH

The unicyclic graph Cn,gobtained by appending a cycle Cgof length g to a pendent vertex of a path on n - g vertices is the lollipop graph on n vertices. In [Algebraic connectivity of lollipop graphs, Linear Algebra Appl.434 (2011) 2204–2210], Guo et al. proved that a( Cn,g-1) < a( Cn,g) for g ≥ 4, where a( Cn,g) is the algebraic connectivity of Cn,g. In this paper, we present a new approach which is quite different from that of Guo et al. in proving a( Cn,g-1) < a( Cn,g) for g ≥ 4.

English

In 1982, Prodinger and Tichy defined the Fibonacci number of a graph G to be the number of independent sets of the graph G. They did so since the Fibonacci number of the path graph Pn is the Fibonacci number F(n+2) and the Fibonacci number of the cycle graph Cn is the Lucas number Ln. The tadpole graph Tn,k is the graph created by concatenating Cn and Pk with an edge from any vertex of Cn to a pendant of Pk for integers n=3 and k=0. This paper establishes formulae and identities for the Fibonacci number of the tadpole graph via algebraic and combinatorial methods.

Locating and identifying codes in circulant networks

A set S of vertices of a graph G is a dominating set of G if every vertex u of G is either in S or it has a neighbour in S. In other words, S is dominating if the sets S∩N[u] where u∈V(G) and N[u] denotes the closed neighbourhood of u in G, are all nonempty. A set S⊆V(G) is called a locating code in G, if the sets S∩N[u] where u∈V(G)∖S are all nonempty and distinct. A set S⊆V(G) is called an identifying code in G, if the sets S∩N[u] where u∈V(G) are all nonempty and distinct. We study locating and identifying codes in the circulant networks Cn(1,3). For an integer n⩾7, the graph Cn(1,3) has vertex set Zn and edges xy where x,y∈Zn and |x−y|∈{1,3}. We prove that a smallest locating code in Cn(1,3) has size ⌈n/3⌉+c, where c∈{0,1}, and a smallest identifying code in Cn(1,3) has size ⌈4n/11⌉+c′, where c′∈{0,1}.

Two Labelings Techniques for Corona Graph Cn*K1

Labeling of graphs is an assignment of integers to the vertices or edges or both using some conditions. There are various labelings techniques such as feliticious labeling, graceful labelings, harmonious labeling, magic labeling etc. applied to certain classes of graphs. H-cordiality is proved for wheels, generalized Peterson graph, triangular graph and special ladder. In this paper, H-cordial labeling and prime labelings of the corona graph Cn*K1 for n ≥ 3 is obtained.

The Odd Harmonious Labeling of Dumbbell and Generalized Prism Graphs

A graph G = (V, E) with |E| = q is said to be odd harmonious if there exists an injection f: V (G) → {0, 1, 2,…, 2q - 1} such that the induced function f*: E(G) → {1, 3, 5,…, 2q - 1} defined by f*(xy) = f(x)+f(y) is a bijection. Then f is said to be odd harmonious labeling of G. A dumbbell graph Dn,k,2 is a bicyclic graph consisting of two vertex-disjoint cycles Cn, Ck and a path P2 joining one vertex of Cn with one vertex of Ck. A prism graph Cn x Pm is a Cartesian product of cycle Cn and path Pm. In this paper we show that the dumbbell graph Dn,k,2 is odd harmonious for n ≡ k ≡ 0 (mod 4) and n ≡ k ≡ 2 (mod 4), generalized prism graph Cn x Pm is odd harmonious for n ≡ 0 (mod 4) and for any m, and generalized prism graph Cn x Pm is not odd harmonious for n ≡ 2 (mod 4).

On Prime Labeling of some Classes of Graphs

Graph labeling is an important area of research in Graph theory. There are many kinds of graph labeling such as Graceful labeling, Magic labeling, Prime labeling, and other different labeling techniques.In this paper the Prime labeling of certain classes of graphs are discussed.It is of interest to note that H-graph which is a 3 –regular graph satisfy Prime labeling. A Gear graph is a graph obtained from Wheel graph, with a vertex added between each pair of adjacent vertices of an outer cycle. It is proved in general this graph is Prime. Yet another class of graphs is the Sun flower graph and corona of Cycle graph Cn and K1,3 .A stepwise algorithm is given to prove that both these classes of graphs satisfy Prime labeling.

On Edge-Balance Index Sets of the Graph C<sub>n</sub>×P<sub>b</sub>(n=0,1,2mod6)

Let G be a simple graph with vertex set V(G) and edge set E(G), and let Z2=(0,1) For a given binary edge labeling f:E(G)→Z2,the edge labeling f induces a partial vertex labeling f*:V(G)→Z2 such that f*(v)=1(0) iff the number of 1-edges (0-edges) is strictly greater than the number of 0-edges (1-edges) incident to , otherwise f*(v) is undefined. For i∈Z2, let v(i)=card(e∈V(G):f*(v)=i) and e(i)=card(e∈E(G):f(e)=i). The edge-balance index sets of a graph G,EBI(G), is defined as {|v(0)-v(1): the edge labeling f satisfies } . In this paper, we completely determine the edge-balance index |e(0)-e(1)|≤1 sets of the graph Cn×Pb(n=0,1,2 mod 6)

Even Vertex Graceful of Path, Circuit, Star, Wheel, some Extension-friendship Graphs and Helm Graph

Even Vertex Graceful of Path, Circuit, Star, Wheel, some Extension-friendship Graphs and Helm Graph

The Merrifield-Simmons indices and Hosoya indices of some classes of cartesian graph product

The Merrifield-Simmons index of a graph is defined as the total number of the independent sets of the graph and the Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we give formula for Merrifield-Simmons and Hosoya indices of some classes of cartesian product of two graphs K2 ◊ H, where H is a path graph Pn, cyclic graph Cn, or star graph Sn, with n vertices (These are called: ladder graph, prism graph, and book graph).

Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

Graphs and Algorithms For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.

Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

Graphs and Algorithms For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.

Choosability of Powers of Circuits

It is proved that ch(G)=χ(G) if G=C n p , the pth power of the circuit graph C n , or if G is a uniform inflation of such a graph. The proof uses the method of Alon and Tarsi. As a corollary, the (a : b)-choosability conjectures hold for all such graphs.

On Characterization of Cyclic Structures

We consider a characterization of cyclic structures based on the information from graph distance matrix, D, and detour matrix, DD, of a graph. Using the elements of D and DD matrices we have constructed a new matrix D/DD, the elements of which are given as a quotient of the corresponding elements of D and DD matrices. The sum of all entries of a matrix above the diagonal of the D/DD matrix shows a regular variation for graphs of the same size with the increase of its cyclic structure. Using the cycle graph Cn and the complete graph Kn as the extreme cases of cyclic structures we propose an index γ as the measure of molecular cyclic structure. The approach is extended to characterization of molecular local features, such as individual rings embedded in different local environment of a molecule.