Degree conditions for the partition of a graph into cycles, edges and isolated vertices

Let k,n be integers with 2≤k≤n, and let G be a graph of order n. We prove that if max{d_G(x),d_G(y)}≥(n−k+1)/2 for any x,y∈V(G) with x≠y and xy∉E(G), then G has k vertex-disjoint subgraphs H₁,…,H_k such that V(H₁)∪?∪V(H_k)=V(G) and H_i is a cycle or K₁ or K₂ for each 1≤i≤k, unless k=2 and G=C₅, or k=3 and G=K₁∪C₅.     

A lower bound for the circumference of a graph

Let G=(V, E) be a block of order n, different from K_n. Let m=min {d(x)+d(y): [x, y]?E}. We show that if m?n then G contains a cycle of length at least m.     

Ordering connected graphs having small degree distances

The concept of degree distance of a connected graph G is a variation of the well-known Wiener index, in which the degrees of vertices are also involved. It is defined by D^′(G)=∑_x∈V(G)d(x)∑_y∈V(G)d(x,y), where d(x) and d(x,y) are the degree of x and the distance between x and y, respectively. In this paper it is proved that connected graphs of order n≥4 having the smallest degree distances are K_1,n−1,BS(n−3,1) and K_1,n−1+e (in this order), where BS(n−3,1) denotes the bistar consisting of vertex disjoint stars K_1,n−3 and K_1,1 with central vertices joined by an edge.     

Computation of Topological Indices of Some Graphs

Let G=(V,E) be a simple connected graph with vertex set V and edge set E. The Wiener index of G is defined by W(G)=∑_{x,y}⊆V d(x,y), where d(x,y) is the length of the shortest path from x to y. The Szeged index of G is defined by Sz(G)=∑ _e=uv∈E n _u (e|G)n _v (e|G), where n _u (e|G) (resp. n _v (e|G)) is the number of vertices of G closer to u (resp. v) than v (resp. u). The Padmakar–Ivan index of G is defined by PI(G)=∑ _e=uv∈E [n _eu (e|G)+n _ev (e|G)], where n _eu (e|G) (resp. n _ev (e|G)) is the number of edges of G closer to u (resp. v) than v (resp. u). In this paper we find the above indices for various graphs using the group of automorphisms of G. This is an efficient method of finding these indices especially when the automorphism group of G has a few orbits on V or E. We also find the Wiener indices of a few graphs which frequently arise in mathematical chemistry using inductive methods.     

On the Existence of a Long Path Between Specified Vertices in a 2-Connected Graph

 Let G be a 2-connected graph with maximum degree Δ (G)≥d, and let x and y be distinct vertices of G. Let W be a subset of V(G)−{x, y} with cardinality at most d−1. Suppose that max{d _G(u), d _G(v)}≥d for every pair of vertices u and v in V(G)−({x, y}∪W) with d _G(u,v)=2. Then x and y are connected by a path of length at least d−|W|. Received: February 5, 1998 Revised: April 13, 1998     

Asymptotic estimates of sums involving the Moebius function

Let p(n) denote the smallest prime factor of an integer n>1 and let p(1)=∞. We study the asymptotic behavior of the sum M(x,y)=Σ_{1≤n≤x,p(n)>y}μ(n) and use this to estimate the size of A(x)=max_|f|≤1|Σ_2≤n<xμ(n)f(p(n))|, where μ(n) is the Moebius function. Applications of bounds for A(x), M(x,y) and similar quantities are discussed.     

Circuits including a given set of vertices

Let G = (V, E) be a digraph of order n, satisfying Woodall's condition ? x, y ∈ V, if (x, y) ? E, then d⁺(x) + d^?(y) ≥ n. Let S be a subset of V of cardinality s. Then there exists a circuit including S and of length at most Min(n, 2s). In the case of oriented graphs we obtain the same result under the weaker condition d⁺(x) + d^?(y) ≥ n – 2 (which implies hamiltonism).     

Binding number and minimum degree for the existence of (g,f,n)-critical graphs

Let G be a graph of order p. The binding number of G is defined as $\mbox{bind}(G):=\min\{\frac{|N_{G}(X)|}{|X|}\mid\emptyset\neq X\subseteq V(G)\,\,\mbox{and}\,\,N_{G}(X)\neq V(G)\}$ . Let g(x) and f(x) be two nonnegative integer-valued functions defined on V(G) with g(x)≤f(x) for any x∈V(G). A graph G is said to be (g,f,n)-critical if G?N has a (g,f)-factor for each N?V(G) with |N|=n. If g(x)≡a and f(x)≡b for all x∈V(G), then a (g,f,n)-critical graph is an (a,b,n)-critical graph. In this paper, several sufficient conditions on binding number and minimum degree for graphs to be (a,b,n)-critical or (g,f,n)-critical are given. Moreover, we show that the results in this paper are best possible in some sense.     

Partial Parity (g,f )-Factors and Subgraphs Covering Given Vertex Subsets

 Let G be a graph and W a subset of V(G). Let g,f:V(G)→Z be two integer-valued functions such that g(x)≤f(x) for all x∈V(G) and g(y)≡f(y) (mod 2) for all y∈W. Then a spanning subgraph F of G is called a partial parity (g,f)-factor with respect to W if g(x)≤deg _F (x)≤f(x) for all x∈V(G) and deg _F (y)≡f(y) (mod 2) for all y∈W. We obtain a criterion for a graph G to have a partial parity (g,f)-factor with respect to W. Furthermore, by making use of this criterion, we give some necessary and sufficient conditions for a graph G to have a subgraph which covers W and has a certain given property. Received: June 14, 1999?Final version received: August 21, 2000     

Vertex-disjoint quadrilaterals containing specified edges in a bipartite graph

The theory of vertex-disjoint cycles and 2-factors of graphs is the extension and generation of the well-known Hamiltonian cycles theory and it has important applications in network communication. In this paper we first prove the following result. Let G=(V ₁,V ₂;E) be a bipartite graph with |V ₁|=|V ₂|=n such that n≥2k+1, where k≥1 is an integer. If d(x)+d(y)≥?(4n+2k?1)/3? for each pair of nonadjacent vertices x and y of G with x∈V ₁ and y∈V ₂, then, for any k independent edges e ₁,…,e _k of G, G contains k vertex-disjoint quadrilaterals C ₁,…,C _k such that e _i ∈E(C _i ) for each i∈{1,…,k}. We further show that the degree condition above is sharp. If |V ₁|=|V ₂|=2k, we give degree conditions that G has a 2-factor with k vertex-disjoint quadrilaterals C ₁,…,C _k containing specified edges of G.     

Distance matrix polynomials of trees

Let G be a finite connected graph. If x and y are vertices of G, one may define a distance function d_G on G by letting d_G(x, y) be the minimal length of any path between x and y in G (with d_G(x, x) = 0). Thus, for example, d_G(x, y) = 1 if and only if {x, y} is an edge of G. Furthermore, we define the distance matrix D(G) for G to be the square matrix with rows and columns indexed by the vertex set of G which has d_G(x, y) as its (x, y) entry. In this paper we are concerned with properties of D(G) for the case in which G is a tree (i.e., G is acyclic). In particular, we precisely determine the coefficients of the characteristic polynomial of D(G). This determination is made by deriving surprisingly simple expressions for these coefficients as certain fixed linear combinations of the numbers of various subgraphs of G.     

Super-vertex-antimagic total labelings of disconnected graphs

Let G=(V,E) be a finite, simple and non-empty (p,q)-graph of order p and size q. An (a,d)-vertex-antimagic total labeling is a bijection f from V(G)∪E(G) onto the set of consecutive integers 1,2,…,p+q, such that the vertex-weights form an arithmetic progression with the initial term a and the common difference d, where the vertex-weight of x is the sum of values f(xy) assigned to all edges xy incident to vertex x together with the value assigned to x itself, i.e. f(x). Such a labeling is called super if the smallest possible labels appear on the vertices.In this paper, we will study the properties of such labelings and examine their existence for disconnected graphs.     

Bounds on the index of the signless Laplacian of a graph

Let G=(V,E) be a simple, undirected graph of order n and size m with vertex set V, edge set E, adjacency matrix A and vertex degrees Δ=d₁≥d₂≥?≥d_n=δ. The average degree of the neighbor of vertex v_i is . Let D be the diagonal matrix of degrees of G. Then L(G)=D(G)−A(G) is the Laplacian matrix of G and Q(G)=D(G)+A(G) the signless Laplacian matrix of G. Let μ₁(G) denote the index of L(G) and q₁(G) the index of Q(G). We survey upper bounds on μ₁(G) and q₁(G) given in terms of the d_i and m_i, as well as the numbers of common neighbors of pairs of vertices. It is well known that μ₁(G)≤q₁(G). We show that many but not all upper bounds on μ₁(G) are still valid for q₁(G).     

Centralizers of generalized derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2, with Utumi quotient ring U and extended centroid C, δ a nonzero derivation of R, G a nonzero generalized derivation of R, and f(x ₁, …, x _n ) a noncentral multilinear polynomial over C. If δ(G(f(r ₁, …, r _n ))f(r ₁, …, r _n )) = 0 for all r ₁, …, r _n ∈ R, then f(x ₁, …, x _n )² is central-valued on R. Moreover there exists a ∈ U such that G(x) = ax for all x ∈ R and δ is an inner derivation of R such that δ(a) = 0.     

Bandwidth of the strong product of two connected graphs

The bandwidth B(G) of a graph G is the minimum of the quantity max{|f(x)-f(y)|:xy∈E(G)} taken over all proper numberings f of G. The strong product of two graphs G and H, written as G(S_P)H, is the graph with vertex set V(G)×V(H) and with (u₁,v₁) adjacent to (u₂,v₂) if one of the following holds: (a) u₁ and v₁ are adjacent to u₂ and v₂ in G and H, respectively, (b) u₁ is adjacent to u₂ in G and v₁=v₂, or (c) u₁=u₂ and v₁ is adjacent to v₂ in H. In this paper, we investigate the bandwidth of the strong product of two connected graphs. Let G be a connected graph. We denote the diameter of G by D(G). Let d be a positive integer and let x,y be two vertices of G. Let denote the set of vertices v so that the distance between x and v in G is at most d. We define δ_d(G) as the minimum value of over all vertices x of G. Let denote the set of vertices z such that the distance between x and z in G is at most d-1 and z is adjacent to y. We denote the larger of and by . We define η(G)=1 if G is complete and η(G) as the minimum of over all pair of vertices x,y of G otherwise. Let G and H be two connected graphs. Among other results, we prove that if δ_D(H)(G)?B(G)D(H)+1 and B(H)=⌈(|V(H)|+η(H)-2)/D(H)⌉, then B(G(S_P)H)=B(G)|V(H)|+B(H). Moreover, we show that this result determines the bandwidth of the strong product of some classes of graphs. Furthermore, we study the bandwidth of the strong product of power of paths with complete bipartite graphs.     

A note on Z 3-connected graphs with degree sum condition

A graph G satisfies the Ore-condition if d(x) + d(y) ≥ | V (G) | for any xy ■ E(G). Luo et al. [European J. Combin., 2008] characterized the simple Z3-connected graphs satisfying the Ore-condition. In this paper, we characterize the simple Z3-connected graphs G satisfying d(x) + d(y) ≥ | V (G) |-1 for any xy ■ E(G), which improves the results of Luo et al.     

Periodic solutions to one-dimensional wave equation with x-dependent coefficients

In this paper, we study the nonlinear one-dimensional periodic wave equation with x-dependent coefficients u(x)ytt−(ux(x)yx)+g(x,t,y)=f(x,t) on (0,π)×R under the boundary conditions a₁y(0,t)+b₁yx(0,t)=0, a₂y(π,t)+b₂yx(π,t)=0 ( for i=1,2) and the periodic conditions y(x,t+T)=y(x,t), yt(x,t+T)=yt(x,t). Such a model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. A main concept is the notion “weak solution” to be given in Section 2. For T is the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.     

On friendly index sets of 2-regular graphs

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∈A, let v_f(i)=card{v∈V:f(v)=i} and e_f(i)=card{e∈E:f^∗(e)=i}. A labeling f is said to be A-friendly if |v_f(i)−v_f(j)|≤1 for all (i,j)∈A×A, and A-cordial if we also have |e_f(i)−e_f(j)|≤1 for all (i,j)∈A×A. When A=Z₂, the friendly index set of the graph G is defined as {|e_f(1)−e_f(0)|:the vertex labelingf is Z₂-friendly}. In this paper we completely determine the friendly index sets of 2-regular graphs. In particular, we show that a 2-regular graph of order n is cordial if and only if n?2 (mod 4).     

T-Colorings and T-Edge Spans of Graphs

 Suppose G is a graph and T is a set of non-negative integers that contains 0. A T-coloring of G is an assignment of a non-negative integer f(x) to each vertex x of G such that |f(x)−f(y)|∉T whenever xy∈E(G). The edge span of a T-coloring−f is the maximum value of |f(x) f(y)| over all edges xy, and the T-edge span of a graph G is the minimum value of the edge span of a T-coloring of G. This paper studies the T-edge span of the dth power C ^d _n of the n-cycle C _n for T={0, 1, 2, …, k−1}. In particular, we find the exact value of the T-edge span of C _n ^d for n≡0 or (mod d+1), and lower and upper bounds for other cases. Received: May 13, 1996 Revised: December 8, 1997