Path Pn Research Articles

Solitaire clobber on circulant graphs

Solitaire Clobber is a one-player combinatorial game on graphs. Each vertex of a graph G starts with a black or a white stone. A stone on one vertex can clobber an adjacent stone of the opposite color, removing it and taking its place. The goal is to minimize the number of stones remaining when no further move is possible. An initial configuration is k-reducible if it can be reduced to k stones. A graph is strongly 1-reducible if, for any vertex v, any initial configuration that is not monochromatic outside v can be reduced to one stone, on v, of either color. Every such graph has a Hamiltonian path ending at v. For the path Pn, we prove that the rth distance power Pnr is strongly 1-reducible when r≥3 but not when r=2 (Pn2 is 2-reducible). As a consequence, circulant graphs containing edges of lengths 1, 2, and 3 are strongly 1-reducible; we show also that those containing Cn2 are 1-reducible.

New Families of Odd Harmonious Graphs

In this paper, we show that the number of edges for any odd harmonious Eulerian graph is congruent to 0 or 2 (mod 4), and we found a counter example for the inverse of this statement is not true. We also proved that, the graphs which are constructed by two copies of even cycle Cn sharing a common edge are odd harmonious. In addition, we obtained an odd harmonious labeling for the graphs which are constructed by two copies of cycle Cn sharing a common vertex when n is congruent to 0 (mod 4). Moreover, we show that, the Cartesian product of cycle graph Cm and path Pn for each ) 4 (mod

Square Graceful Graphs

A ( p, q) graph G(V, E) is said to be a square graceful graph if there exists an injection f :V(G)---{0,1,2,3,..., q2 } such that the induced mapping f p : E(G) --- {1, 4, 9,..., q2} defined by f (uv)=| f (u) f (v)| p = ? is a bijection. The function f is called a square labeling of G . In this paper, we prove that the star n K 1, , bistar m n B , , the graph obtained by the subdivision of the edges of the star K1,n , the graph obtained by the subdivision of the central edge of the bistar Bm,n , the generalised crown n C K 3 1, ? , graph 1 P nK m ? (n ? 2) , the comb 1 P K n? , graph ( , ) m n P S , (3, t) kite graph (t ? 2) and the path Pn are square graceful graphs.

On path-Jahangir Ramsey numbers

For given graphs G and H, the Ramsey number R(G,H) is the smallest positive integer N such that for every graph F of order N the following holds: either F contains G as a subgraph or the complement of F contains H as a subgraph. For the path Pn and the Jahangir graph Jm, it is proved that R(P4, J4) = 6, R(Pn, J4) = n + 1 for n ≥ 5 and R(Pn, Jm) = n + m 2 − 1 for even m ≥ 6 and n ≥ (2m − 1)(m2 − 1) + 1. We also suggest an open problem. Mathematics Subject Classifications: 05C55, 05D10

On the crossing numbers of Cartesian products of paths with special graphs

There are known exact results of the crossing numbers of the Cartesian product of all graphs of order at most four with paths, cycles and stars. Moreover, for the path Pn of length n, the crossing numbers of Cartesian products GPn for all connected graphs G on five vertices and for forty graphs G on six vertices are known. In this paper, we extend these results by determining the crossing numbers of the Cartesian products GPn for six other graphs G of order six.

Some New Results on Distance k-Domination in Graphs

We determine the distance k-domination number for the total graph, shadow graph, and middle graph of path Pn.

Signed 2-independence of Cartesian product of directed paths

A two-valued function f: V(D)→{−1, 1} defined on the vertices of a digraph D=(V(D), A(D)) is called a signed 2-independence function (S2IF) if f(N−[v])≤1 for every v in D. The weight of a S2IF is f(V(D))=∑v∈V(D)f(v). The maximum weight of a S2IF of D is the signed 2-independence number (or the lower against number) of D. Let Pm×Pn be the Cartesian product of directed paths Pm and Pn. In this paper, we determine the exact values of for 1≤m≤5 and n≥1.

Edge Pair Sum 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*(?) = ?e?E? f(e) is one-one, where denotes the set of edges in G that are incident with a vertex v and f*(V(G)) is either of the form {±k1, ±k2, , ±kp/2} or {±k1, ±k2, , ±k(p-1)/2}U{kp/2}according as p is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper we prove that path Pn, cycle Cn, triangular snake, PmUK1,n, Cn?Kmc are edge pair sum graphs. Keywords: Pair sum graph, edge pair sum labeling, edge pair sum graph.© 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi: http://dx.doi.org/10.3329/jsr.v5i3.15001 J. Sci. Res. 5 (3), 457-467 (2013)

On the signless Laplacian coefficients of unicyclic graphs

Let G be a graph of order n and let QG(x)=∑i=0n(−1)ipi(G)xn−i be the characteristic polynomial of the signless Laplacian of G. Let Eg,n (respectively, Cg(Sn−g+1)) denote the unicyclic graph of order n obtained by a coalescence of a vertex in the cycle Cg with an end vertex (respectively, the center) of the path Pn−g+1 (respectively, the star Sn−g+1). It is proved that for k=2,…,n−1, as G varies over all unicyclic graphs of order n, depending on k and n, the maximum value of pk(G) is attained at G=Cn or E3,n, and the minimum value is attained uniquely at G=C4(Sn−3) or C3(Sn−2). Except for the resolution of a conjecture on cubic polynomials, the uniqueness issue for the maximization problem is also settled.

On square sum graphs

A (p, q)-graph G is said to be square sum, if there exists a bijection f : V(G) → {0,1, 2,...,p — 1} such that the induced function f * : E(G) → N given by f * (uv) = (f (u))2 + (f (v))2 for every uv ∈ E(G) is injective. In this paper we initiate a study on square sum graphs and prove that trees, unicyclic graphs, mCn, m > 1, cycle with a chord, the graph obtained by joining two copies of cycle Cn by a path Pk and the graph defined by path union of k copies of Cn, when the path Pn = P2 are square sum.

The crossing numbers of products of path with graphs of order six

The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. For the path Pn of length n, the crossing numbers of Cartesian products G Pn for all connected graphs G on five vertices are also known. In this paper, the crossing numbers of Cartesian products G Pn for graphs G of order six are studied. Let H denote the unique tree of order six with two vertices of degree three. The main contribution is that the crossing number of the Cartesian product H Pn is 2(n − 1). In addition, the crossing numbers of G Pn for fourty graphs G on six vertices are collected.

Pretty good state transfer on double stars

Let A be the adjacency matrix of a graph X and suppose U(t)=exp(itA). We view A as acting on CxV(X) and take the standard basis of this space to be the vectors eu for u in V(X). hysicists say that we have perfect state transfer from vertex u to v at time τ if there is a scalar γ such that U(τ)eu=γev.(Since U(t) is unitary, ‖γ=1‖.) For example, if X is the d-cube and u and v are at distance d then we have perfect state transfer from u to v at time π/2. Despite the existence of this nice family, it has become clear that perfect state transfer is rare. Hence we consider a relaxation: we say that we have pretty good state transfer from u to v if there is a complex number γ and, for each positive real ϵ there is a time t such that ‖U(t)eu-γev‖<ϵ.Again we necessarily have |γ|=1.In a recent paper Godsil, Kirkland, Severini and Smith showed that we have have pretty good state transfer between the end vertices of the path Pn if and only n+1 is a power of two, a prime, or twice a prime. (There is perfect state transfer between the end vertices only for P2 and P3.) It is something of a surprise that the occurrence of pretty good state transfer is characterized by a number-theoretic condition. In this paper we extend the theory of pretty good state transfer. We provide what is only the second family of graphs where pretty good state transfer occurs. The graphs we use are the double-star graphs Sk,ℓ, these are trees with a vertex of degree k+1 adjacent to a vertex of degree ℓ+1, and all other vertices of degree one. We prove that perfect state transfer does not occur in any graph in this family. We show that if ℓ>2, then there is pretty good state transfer in S2,ℓ between the two end vertices adjacent to the vertex of degree three. If k,ℓ>2, we prove that there is never perfect state transfer between the two vertices of degree at least three, and we show that there is pretty good state transfer between them if and only these vertices both have degree k+1 and 4k+1 is not a perfect square. Thus we find again the the existence of perfect state transfer depends on a number theoretic condition. It is also interesting that although no double stars have perfect state transfer, there are some that admit pretty good state transfer.

The weighted vertex PI index

The vertex PI index is a distance-based molecular structure descriptor, that recently found numerous chemical applications. In order to increase diversity of this topological index for bipartite graphs, we introduce a weighted version defined as PIw(G)=∑e=uv∈E(deg(u)+deg(v))(nu(e)+nv(e)), where deg(u) denotes the vertex degree of u and nu(e) denotes the number of vertices of G whose distance to the vertex u is smaller than the distance to the vertex v. We establish basic properties of PIw(G), and prove various lower and upper bounds. In particular, the path Pn has minimal, while the complete tripartite graph Kn/3,n/3,n/3 has maximal weighed vertex PI index among connected graphs with n vertices. We also compute exact expressions for the weighted vertex PI index of the Cartesian product of graphs. Finally we present modifications of two inequalities and open new perspectives for the future research.

The total bondage number of grid graphs

The total domination number of a graph G without isolated vertices is the minimum number of vertices that dominate all vertices in G. The total bondage number bt(G) of G is the minimum number of edges whose removal enlarges the total domination number. This paper considers grid graphs. An (n,m)-grid graph Gn,m is defined as the cartesian product of two paths Pn and Pm. This paper determines the exact values of bt(Gn,2) and bt(Gn,3), and establishes some upper bounds of bt(Gn,4).

Permutation Labeling for some Shadow Graphs

A permutation labeling of a graph G is a bijective assignment of labels from { 1,2,3, …p} to the vertices of G such that when each edge of G has assigned a weight defined by the number of permutations of f(u) things taken f(v) at a time. Such a labeling f is called permutation labeling of G. A graph which admits permutation labeling is called permutation graphs. In this paper I proved that the shadow graphs of path Pn, star K1,n and path union of shadow graphs of cycle Cn are permutation graphs. Further I proved that the split graphs of path Pn and star K1,n are permutation graphs.

Square Sum Labeling For Some Middle and Total Graphs

A (p, q) graph G is said to be a square sum graph if there exist a bijection f: V(G)→{0,1,2,….p-1} such that the induced function f * :E(G) → N given by f * (u v) =[f * (u)] 2 +[f * (v)] 2 for every uv ∈ E(G) are all distinct. In this paper the square sum labeling of total graph of path Pn, cycle Cn and middle graph of path Pn , cycle Cn are discussed.

Spectra of Manhattan Products of Directed Paths Pn#P2

Spectra of Manhattan Products of Directed Paths Pn#P2

Bounds for the Kirchhoff Index of Bipartite Graphs

A (m, n)‐bipartite graph is a bipartite graph such that one bipartition has m vertices and the other bipartition has n vertices. The tree dumbbell D(n, a, b) consists of the path Pn−a−b together with a independent vertices adjacent to one pendent vertex of Pn−a−b and b independent vertices adjacent to the other pendent vertex of Pn−a−b. In this paper, firstly, we show that, among (m, n)‐bipartite graphs (m ≤ n), the complete bipartite graph Km,n has minimal Kirchhoff index and the tree dumbbell D(m + n, ⌊n − (m + 1)/2⌋, ⌈n − (m + 1)/2⌉) has maximal Kirchhoff index. Then, we show that, among all bipartite graphs of order l, the complete bipartite graph K⌊l/2⌋,l−⌊l/2⌋ has minimal Kirchhoff index and the path Pl has maximal Kirchhoff index, respectively. Finally, bonds for the Kirchhoff index of (m, n)‐bipartite graphs and bipartite graphs of order l are obtained by computing the Kirchhoff index of these extremal graphs.

Spectral determination of some chemical graphs

Let Tk denote the caterpillar obtained by attaching k pendant edges at two pendant vertices of the path Pn and two pendant edges at the other vertices of Pn. It is proved that Tk is determined by its signless Laplacian spectrum when k = 2 or 3, while T2 by its Laplacian spectrum.

Vertex-Distinguishing E-Total Colorings of Graphs

Let G be a simple graph. A total coloring f of G is called an E-total coloring if no two adjacent vertices of G receive the same color, and no edge of G receives the same color as one of its endpoints. For an E-total coloring f of a graph G and any vertex u of G let Cf(u) or C(u) denote the set of colors of vertex u and of the edges incident to u. We call C(u) the color set of u. If C(u) ≠ C(v) for any two different vertices u and v of V(G) then we say that f is a vertex-distinguishing E-total coloring of G or a VDET coloring of G for short. The minimum number of colors required for a VDET coloring of G is denoted by \({\chi_{vt}^e(G)}\) and is called the VDET chromatic number of G. In this paper, we find the VDET chromatic number for some special families of graphs, such as the path Pn, the cycle Cn, the complete bipartite graphs K1, n and K2, n the complete graph Kn, and wheels and fans, and we also propose a related conjecture.