Geodetic and Steiner geodetic sets in 3-Steiner distance hereditary graphs

Let G be a connected graph and S⊆V(G). Then the Steiner distance of S, denoted by d_G(S), is the smallest number of edges in a connected subgraph of G containing S. Such a subgraph is necessarily a tree called a Steiner tree for S. The Steiner interval for a set S of vertices in a graph, denoted by I(S) is the union of all vertices that belong to some Steiner tree for S. If S={u,v}, then I(S) is the interval I[u,v] between u and v. A connected graph G is 3-Steiner distance hereditary (3-SDH) if, for every connected induced subgraph H of order at least 3 and every set S of three vertices of H, d_H(S)=d_G(S). The eccentricity of a vertex v in a connected graph G is defined as e(v)=max{d(v,x)|x∈V(G)}. A vertex v in a graph G is a contour vertex if for every vertex u adjacent with v, e(u)?e(v). The closure of a set S of vertices, denoted by I[S], is defined to be the union of intervals between pairs of vertices of S taken over all pairs of vertices in S. A set of vertices of a graph G is a geodetic set if its closure is the vertex set of G. The smallest cardinality of a geodetic set of G is called the geodetic number of G and is denoted by g(G). A set S of vertices of a connected graph G is a Steiner geodetic set for G if I(S)=V(G). The smallest cardinality of a Steiner geodetic set of G is called the Steiner geodetic number of G and is denoted by sg(G). We show that the contour vertices of 3-SDH and HHD-free graphs are geodetic sets. For 3-SDH graphs we also show that g(G)?sg(G). An efficient algorithm for finding Steiner intervals in 3-SDH graphs is developed.     

Steiner intervals and Steiner geodetic numbers in distance-hereditary graphs

A Steiner tree for a set S of vertices in a connected graph G is a connected subgraph of G with a smallest number of edges that contains S. The Steiner interval I(S) of S is the union of all the vertices of G that belong to some Steiner tree for S. If S={u,v}, then I(S)=I[u,v] is called the interval between u and v and consists of all vertices that lie on some shortest u-v path in G. The smallest cardinality of a set S of vertices such that ?_u,v∈SI[u,v]=V(G) is called the geodetic number and is denoted by g(G). The smallest cardinality of a set S of vertices of G such that I(S)=V(G) is called the Steiner geodetic number of G and is denoted by sg(G). We show that for distance-hereditary graphs g(G)?sg(G) but that g(G)/sg(G) can be arbitrarily large if G is not distance hereditary. An efficient algorithm for finding the Steiner interval for a set of vertices in a distance-hereditary graph is described and it is shown how contour vertices can be used in developing an efficient algorithm for finding the Steiner geodetic number of a distance-hereditary graph.     

Subpancyclicity of line graphs and degree sums along paths

A graph is called subpancyclic if it contains a cycle of length ? for each ? between 3 and the circumference of the graph. We show that if G is a connected graph on n?146 vertices such that d(u)+d(v)+d(x)+d(y)>(n+10/2) for all four vertices u,v,x,y of any path P=uvxy in G, then the line graph L(G) is subpancyclic, unless G is isomorphic to an exceptional graph. Moreover, we show that this result is best possible, even under the assumption that L(G) is hamiltonian. This improves earlier sufficient conditions by a multiplicative factor rather than an additive constant.     

Greedy F-colorings of graphs

Let G=(V,E) be a connected graph. For a symmetric, integer-valued function δ on V×V, where K is an integer constant, N₀ is the set of nonnegative integers, and Z is the set of integers, we define a C-mapping by F(u,v,m)=δ(u,v)+m−K. A coloring c of G is an F-coloring if F(u,v,|c(u)−c(v)|)?0 for every two distinct vertices u and v of G. The maximum color assigned by c to a vertex of G is the value of c, and the F-chromatic number F(G) is the minimum value among all F-colorings of G. For an ordering of the vertices of G, a greedy F-coloring c of s is defined by (1) c(v₁)=1 and (2) for each i with 1?i<n, c(v_i+1) is the smallest positive integer p such that F(v_j,v_i+1,|c(v_j)−p|)?0, for each j with 1?j?i. The greedy F-chromatic number gF(s) of s is the maximum color assigned by c to a vertex of G. The greedy F-chromatic number of G is gF(G)=min{gF(s)} over all orderings s of V. The Grundy F-chromatic number is GF(G)=max{gF(s)} over all orderings s of V. It is shown that gF(G)=F(G) for every graph G and every F-coloring defined on G. The parameters gF(G) and GF(G) are studied and compared for a special case of the C-mapping F on a connected graph G, where δ(u,v) is the distance between u and v and .     

Geodesic-pancyclic graphs

A shortest path connecting two vertices u and v is called a u-v geodesic. The distance between u and v in a graph G, denoted by d_G(u,v), is the number of edges in a u-v geodesic. A graph G with n vertices is panconnected if, for each pair of vertices u,v∈V(G) and for each integer k with d_G(u,v)?k?n-1, there is a path of length k in G that connects u and v. A graph G with n vertices is geodesic-pancyclic if, for each pair of vertices u,v∈V(G), every u-v geodesic lies on every cycle of length k satisfying max{2d_G(u,v),3}?k?n. In this paper, we study sufficient conditions of geodesic-pancyclic graphs. In particular, we show that most of the known sufficient conditions of panconnected graphs can be applied to geodesic-pancyclic graphs.     

The partition dimension of circulant graphs

Let Π = {S₁, S₂, . . . , S_k} be an ordered partition of the vertex set V (G) of a graph G. The partition representation of a vertex v ∈ V (G) with respect to Π is the k-tuple r(v|Π) = (d(v, S₁), d(v, S₂), . . . , d(v, S_k)), where d(v, S) is the distance between v and a set S. If for every pair of distinct vertices u, v ∈ V (G), we have r(u|Π) ≠ r(v|Π), then Π is a resolving partition and the minimum cardinality of a resolving partition of V (G) is called the partition dimension of G. We study the partition dimension of circulant graphs, which are Cayley graphs of cyclic groups. Grigorious et al. [On the partition dimension of circulant graphs] proved that pd(C_n(1, 2, . . . , t)) ≥ t + 1 for n ≥ 3. We disprove this statement by showing that if t ≥ 4 is even, then there exists an infinite set of values of n, such that . We also present exact values of the partition dimension of circulant graphs with 3 generators.     

On the Szeged and the Laplacian Szeged spectrum of a graph

For a given graph G its Szeged weighting is defined by w(e)=n_u(e)n_v(e), where e=uv is an edge of G,n_u(e) is the number of vertices of G closer to u than to v, and n_v(e) is defined analogously. The adjacency matrix of a graph weighted in this way is called its Szeged matrix. In this paper we determine the spectra of Szeged matrices and their Laplacians for several families of graphs. We also present sharp upper and lower bounds on the eigenvalues of Szeged matrices of graphs.     

Sharp bounds for the largest eigenvalue of the signless Laplacian of a graph

Let G be a simple connected graph with n vertices and m edges. Denote the degree of vertex v_i by d(v_i). The matrix Q(G)=D(G)+A(G) is called the signless Laplacian of G, where D(G)=diag(d(v₁),d(v₂),…,d(v_n)) and A(G) denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Let q₁(G) be the largest eigenvalue of Q(G). In this paper, we first present two sharp upper bounds for q₁(G) involving the maximum degree and the minimum degree of the vertices of G and give a new proving method on another sharp upper bound for q₁(G). Then we present three sharp lower bounds for q₁(G) involving the maximum degree and the minimum degree of the vertices of G. Moreover, we determine all extremal graphs which attain these sharp bounds.     

Local-edge-connectivity in digraphs and oriented graphs

A digraph without any cycle of length two is called an oriented graph. The local-edge-connectivityλ(u,v) of two vertices u and v in a digraph or graph D is the maximum number of edge-disjoint u-v paths in D, and the edge-connectivity of D is defined as . Clearly, λ(u,v)?min{d⁺(u),d^-(v)} for all pairs u and v of vertices in D. Let δ(D) be the minimum degree of D. We call a graph or digraph D maximally edge-connected when λ(D)=δ(D) and maximally local-edge-connected when

λ(u,v)=min{d⁺(u),d^-(v)}     

Degree Distance of Unicyclic Graphs with Given Matching Number

Let G be a connected graph with vertex set V(G). The degree distance of G is defined as ${D'(G) = \sum_{\{u, v\}\subseteq V(G)} (d_G(u) + d_G (v))\, d(u,v)}$ , where d _G (u) is the degree of vertex u, d(u, v) denotes the distance between u and v, and the summation goes over all pairs of vertices in G. In this paper, we characterize n-vertex unicyclic graphs with given matching number and minimal degree distance.     

A note on the weakly convex and convex domination numbers of a torus

The distanced_G(u,v) between two vertices u and v in a connected graph G is the length of the shortest (u,v) path in G. A (u,v) path of length d_G(u,v) is called a (u,v)-geodesic. A set X⊆V is called weakly convex in G if for every two vertices a,b∈X, exists an (a,b)-geodesic, all of whose vertices belong to X. A set X is convex in G if for all a,b∈X all vertices from every (a,b)-geodesic belong to X. The weakly convex domination number of a graph G is the minimum cardinality of a weakly convex dominating set of G, while the convex domination number of a graph G is the minimum cardinality of a convex dominating set of G. In this paper we consider weakly convex and convex domination numbers of tori.     

Upper minus total domination in small-degree regular graphs

A function f:V(G)→{-1,0,1} defined on the vertices of a graph G is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. An MTDF f is minimal if there does not exist an MTDF g:V(G)→{-1,0,1}, f≠g, for which g(v)?f(v) for every v∈V(G). The weight of an MTDF is the sum of its function values over all vertices. The minus total domination number of G is the minimum weight of an MTDF on G, while the upper minus domination number of G is the maximum weight of a minimal MTDF on G. In this paper we present upper bounds on the upper minus total domination number of a cubic graph and a 4-regular graph and characterize the regular graphs attaining these upper bounds.     

The eccentric connectivity index of nanotubes and nanotori

Let G be a molecular graph. The eccentric connectivity index ξ^c(G) is defined as ξ^c(G)=∑_u∈V(G)deg_G(u)ε_G(u), where deg_G(u) denotes the degree of vertex u and ε_G(u) is the largest distance between u and any other vertex v of G. In this paper exact formulas for the eccentric connectivity index of TUC₄C₈(S) nanotube and TC₄C₈(S) nanotorus are given.     

Graph Connectivity After Path Removal

Let G be a graph and u, v be two distinct vertices of G. A u—v path P is called nonseparating if G—V(P) is connected. The purpose of this paper is to study the number of nonseparating u—v path for two arbitrary vertices u and v of a given graph. For a positive integer k, we will show that there is a minimum integer (k) so that if G is an (k)-connected graph and u and v are two arbitrary vertices in G, then there exist k vertex disjoint paths P ₁[u,v], P ₂[u,v], . . ., P _k [u,v], such that G—V (P _i [u,v]) is connected for every i (i = 1, 2, ..., k). In fact, we will prove that (k) 22k+2. It is known that (1) = 3.. A result of Tutte showed that (2) = 3. We show that (3) = 6. In addition, we prove that if G is a 5-connected graph, then for every pair of vertices u and v there exists a path P[u, v] such that G—V(P[u, v]) is 2-connected.* Supported by NSF grant No. DMS-0070059 Supported by ONR grant N00014-97-1-0499 Supported by NSF grant No. 9531824     

A sufficient condition for pancyclability of graphs

Let G be a graph of order n and S be a vertex set of q vertices. We call G,S-pancyclable, if for every integer i with 3≤i≤q there exists a cycle C in G such that |V(C)∩S|=i. For any two nonadjacent vertices u,v of S, we say that u,v are of distance two in S, denoted by d_S(u,v)=2, if there is a path P in G connecting u and v such that |V(P)∩S|≤3. In this paper, we will prove that if G is 2-connected and for all pairs of vertices u,v of S with d_S(u,v)=2, , then there is a cycle in G containing all the vertices of S. Furthermore, if for all pairs of vertices u,v of S with d_S(u,v)=2, , then G is S-pancyclable unless the subgraph induced by S is in a class of special graphs. This generalizes a result of Fan [G. Fan, New sufficient conditions for cycles in graphs, J. Combin. Theory B 37 (1984) 221-227] for the case when S=V(G).     

Unoriented Laplacian maximizing graphs are degree maximal

A connected graph is said to be unoriented Laplacian maximizing if the spectral radius of its unoriented Laplacian matrix attains the maximum among all connected graphs with the same number of vertices and the same number of edges. A graph is said to be threshold (maximal) if its degree sequence is not majorized by the degree sequence of any other graph (and, in addition, the graph is connected). It is proved that an unoriented Laplacian maximizing graph is maximal and also that there are precisely two unoriented Laplacian maximizing graphs of a given order and with nullity 3. Our treatment depends on the following known characterization: a graph G is threshold (maximal) if and only if for every pair of vertices u,v of G, the sets N(u)?{v},N(v)?{u}, where N(u) denotes the neighbor set of u in G, are comparable with respect to the inclusion relation (and, in addition, the graph is connected). A conjecture about graphs that maximize the unoriented Laplacian matrix among all graphs with the same number of vertices and the same number of edges is also posed.     

New upper bounds on the spectral radius of unicyclic graphs

Let G=(V(G),E(G)) be a unicyclic simple undirected graph with largest vertex degree Δ. Let C_r be the unique cycle of G. The graph G-E(C_r) is a forest of r rooted trees T₁,T₂,…,T_r with root vertices v₁,v₂,…,v_r, respectively. Let

     

A short proof of Fan's theorem

In this note, we give a new short proof of the following theorem: Let G be a 2-connected graph of order n. If for any two vertices u and v with d(u,v)=2,max{d(u),d(v)}?c/2, then the circumference of G is at least c, where 3?c?n and d(u,v) is the distance between u and v in G.     

A Greedy Partition Lemma for directed domination

A directed dominating set in a directed graph D is a set S of vertices of V such that every vertex u∈V(D)?S has an adjacent vertex v in S with v directed to u. The directed domination number of D, denoted by γ(D), is the minimum cardinality of a directed dominating set in D. The directed domination number of a graph G, denoted Γ_d(G), is the maximum directed domination number γ(D) over all orientations D of G. The directed domination number of a complete graph was first studied by Erd?s [P. Erd?s On a problem in graph theory, Math. Gaz. 47 (1963) 220–222], albeit in a disguised form. In this paper we prove a Greedy Partition Lemma for directed domination in oriented graphs. Applying this lemma, we obtain bounds on the directed domination number. In particular, if α denotes the independence number of a graph G, we show that α≤Γ_d(G)≤α(1+2ln(n/α)).     

Some sufficient conditions for Hamiltonian property in terms of Wiener-type invariants

The Wiener-type invariants of a simple connected graph G = (V, E) can be expressed in terms of the quantities \(W_{f}=\sum_{\{u,v\}\subseteq V}f(d_{G}(u,v))\) for various choices of the function f(x), where d_G(u,v) is the distance between vertices u and v in G. In this paper, we give some sufficient conditions for a connected graph to be Hamiltonian, a connected graph to be traceable, and a connected bipartite graph to be Hamiltonian in terms of the Wiener-type invariants.