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.     

A sufficient degree condition for a graph to contain all trees of size <Emphasis Type="Italic">k</Emphasis>

The Erdős-Sós conjecture says that a graph G on n vertices and number of edges e(G) > n(k− 1)/2 contains all trees of size k. In this paper we prove a sufficient condition for a graph to contain every tree of size k formulated in terms of the minimum edge degree ζ(G) of a graph G defined as ζ(G) = min{d(u) + d(v) − 2: uv ∈ E(G)}. More precisely, we show that a connected graph G with maximum degree Δ(G) ≥ k and minimum edge degree ζ(G) ≥ 2k − 4 contains every tree of k edges if d _G (x) + d _G (y) ≥ 2k − 4 for all pairs x, y of nonadjacent neighbors of a vertex u of d _G (u) ≥ k.     

A new sufficient condition for Hamiltonian graphs

The study of Hamiltonian graphs began with Dirac’s classic result in 1952. This was followed by that of Ore in 1960. In 1984 Fan generalized both these results with the following result: If G is a 2-connected graph of order n and max{d(u),d(v)}≥n/2 for each pair of vertices u and v with distance d(u,v)=2, then G is Hamiltonian. In 1991 Faudree–Gould–Jacobson–Lesnick proved that if G is a 2-connected graph and |N(u)∪N(v)|+δ(G)≥n for each pair of nonadjacent vertices u,v∈V(G), then G is Hamiltonian. This paper generalizes the above results when G is 3-connected. We show that if G is a 3-connected graph of order n and max{|N(x)∪N(y)|+d(u),|N(w)∪N(z)|+d(v)}≥n for every choice of vertices x,y,u,w,z,v such that d(x,y)=d(y,u)=d(w,z)=d(z,v)=d(u,v)=2 and where x,y and u are three distinct vertices and w,z and v are also three distinct vertices (and possibly |{x,y}∩{w,z}| is 1 or 2), then G is Hamiltonian.     

The eavesdropping number of a graph

Let G be a connected, undirected graph without loops and without multiple edges. For a pair of distinct vertices u and v, a minimum {u, v}-separating set is a smallest set of edges in G whose removal disconnects u and v. The edge connectivity of G, denoted λ(G), is defined to be the minimum cardinality of a minimum {u, v}-separating set as u and v range over all pairs of distinct vertices in G. We introduce and investigate the eavesdropping number, denoted ε(G), which is defined to be the maximum cardinality of a minimum {u, v}-separating set as u and v range over all pairs of distinct vertices in G. Results are presented for regular graphs and maximally locally connected graphs, as well as for a number of common families of graphs.     

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.     

Up-embeddability via girth and the degree-sum of adjacent vertices

Let G be a simple graph of order n and girth g. For any two adjacent vertices u and v of G, if d _G (u) + d _G (v) ⩾ n − 2g + 5 then G is up-embeddable. In the case of 2-edge-connected (resp. 3-edge-connected) graph, G is up-embeddable if d _G (u) + d _G (v) ⩾ n − 2g + 3 (resp. d _G (u) + d _G (v) ⩾ n − 2g −5) for any two adjacent vertices u and v of G. Furthermore, the above three lower bounds are all shown to be tight. This work was supported by National Natural Science Foundation of China (Grant No. 10571013)     

Up-Embeddability of a Graph by Order and Girth

Let G be a connected graph of order n and girth g. If d_G(u) + d_G(v) ≥ n − 2g + 5 for any two non-adjacent vertices u and v, then G is up-embeddable. Further more, the lower bound is best possible. Similarly the result of k-edge connected simple graph with girth g is also obtained, k = 2,3. Partially supported by the Postdoctoral Seience Foundation of Central South University and NNSFC under Grant No. 10751013.     

Edge disjoint Hamilton cycles in graphs

Let G be a graph of order n and k ≥ 0 an integer. It is conjectured in [8] that if for any two vertices u and v of a 2(k + 1)‐connected graph G,d _G (u,v) = 2 implies that max{d(u;G), d(v;G)} ≥ (n/2) + 2k, then G has k + 1 edge disjoint Hamilton cycles. This conjecture is true for k = 0, 1 (see cf. [3] and [8]). It will be proved in this paper that the conjecture is true for every integer k ≥ 0. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 8–20, 2000     

On the maximal eccentric connectivity indices of graphs

For a connected simple graph G, the eccentricity ec(v) of a vertex v in G is the distance from v to a vertex farthest from v, and d(v) denotes the degree of a vertex v. The eccentric connectivity index of G, denoted by ξc(G), is defined as v∈V(G)d(v)ec(v). In this paper, we will determine the graphs with maximal eccentric connectivity index among the connected graphs with n vertices and m edges(n ≤ m ≤ n + 4), and propose a conjecture on the graphs with maximal eccentric connectivity index among the connected graphs with n vertices and m edges(m ≥ n + 5).     

The Convexity Number of a Graph

 For two vertices u and v of a connected graph G, the set I[u,v] consists of all those vertices lying on a u−v shortest path in G, while for a set S of vertices of G, the set I[S] is the union of all sets I[u,v] for u,v∈S. A set S is convex if I[S]=S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. The clique number ω(G) is the maximum cardinality of a clique in G. If G is a connected graph of order n that is not complete, then n≥3 and 2≤ω(G)≤con(G)≤n−1. It is shown that for every triple l,k,n of integers with n≥3 and 2≤l≤k≤n−1, there exists a noncomplete connected graph G of order n with ω(G)=l and con(G)=k. Other results on convex numbers are also presented. Received: August 19, 1998 Final version received: May 17, 2000     

Lollipop graphs are extremal for commute times

Consider a simple random walk on a connected graph G=(V, E). Let C(u, v) be the expected time taken for the walk starting at vertex u to reach vertex v and then go back to u again, i.e., the commute time for u and v, and let C(G)=max_{u, v∈V}C(u, v). Further, let 𝒢(n, m) be the family of connected graphs on n vertices with m edges, , and let 𝒢(n)=∪_m𝒢(n, m) be the family of all connected n‐vertex graphs. It is proved that if G∈(n, m) is such that C(G)=max_{H∈𝒢(n, m)}C(H) then G is either a lollipop graph or a so‐called double‐handled lollipop graph. It is further shown, using this result, that if C(G)=max_H∈𝒢(n)C(H) then G is the full lollipop graph or a full double‐handled lollipop graph with [(2n−1)/3] vertices in the clique unless n≤9 in which case G is the n‐path. ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 131–142, 2000     

Graphs with given number of cut vertices and extremal Merrifield-Simmons index

The Merrifield-Simmons index of a graph is defined as the total number of its independent sets, including the empty set. Denote by G(n,k) the set of connected graphs with n vertices and k cut vertices. In this paper, we characterize the graphs with the maximum and minimum Merrifield-Simmons index, respectively, among all graphs in G(n,k) for all possible k values.     

Even graphs

A nontrivial connected graph G is called even if for each vertex v of G there is a unique vertex v such that d(v, v ) = diam G. Special classes of even graphs are defined and compared to each other. In particular, an even graph G is called symmetric if d(u, v) + d(u, v ) = diam G for all u, v ∈ V(G). Several properties of even and symmetric even graphs are stated. For an even graph of order n and diameter d other than an even cycle it is shown that n ≥ 3d – 1 and conjectured that n ≥ 4d – 4. This conjecture is proved for symmetric even graphs and it is shown that for each pair of integers n, d with n even, d ≥ 2 and n ≥ 4d – 4 there exists an even graph of order n and diameter d. Several ways of constructing new even graphs from known ones are presented.     

Closures,cycles, and paths

In 1960 Ore proved the following theorem: Let G be a graph of order n. If d(u) + d(v)≥n for every pair of nonadjacent vertices u and v, then G is hamiltonian. Since then for several other graph properties similar sufficient degree conditions have been obtained, so‐called “Ore‐type degree conditions”. In [R. J. Faudree, R. H. Schelp, A. Saito, and I. Schiermeyer, Discrete Math 307 (2007), 873–877], Faudree et al. strengthened Ore's theorem as follows: They determined the maximum number of pairs of nonadjacent vertices that can have degree sum less than n (i.e. violate Ore's condition) but still imply that the graph is hamiltonian. In this article we prove that for some other graph properties the corresponding Ore‐type degree conditions can be strengthened as well. These graph properties include traceable graphs, hamiltonian‐connected graphs, k‐leaf‐connected graphs, pancyclic graphs, and graphs having a 2‐factor with two components. Graph closures are computed to show these results. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 314–323, 2012     

Pan-connectedness of graphs with large neighborhood unions

Let G be a simple graph with n vertices. For any v ? V(G){v \in V(G)} , let N(v)={u ? V(G): uv ? E(G)}{N(v)=\{u \in V(G): uv \in E(G)\}} , NC(G) = min{|N(u) èN(v)|: u, v ? V(G){NC(G)= \min \{|N(u) \cup N(v)|: u, v \in V(G)} and uv \not ? E(G)}{uv \not \in E(G)\}} , and NC₂(G) = min{|N(u) èN(v)|: u, v ? V(G){NC_2(G)= \min\{|N(u) \cup N(v)|: u, v \in V(G)} and u and v has distance 2 in E(G)}. Let l ≥ 1 be an integer. A graph G on n ≥ l vertices is [l, n]-pan-connected if for any u, v ? V(G){u, v \in V(G)} , and any integer m with l ≤ m ≤ n, G has a (u, v)-path of length m. In 1998, Wei and Zhu (Graphs Combinatorics 14:263–274, 1998) proved that for a three-connected graph on n ≥ 7 vertices, if NC(G) ≥ n − δ(G) + 1, then G is [6, n]-pan-connected. They conjectured that such graphs should be [5, n]-pan-connected. In this paper, we prove that for a three-connected graph on n ≥ 7 vertices, if NC ₂(G) ≥ n − δ(G) + 1, then G is [5, n]-pan-connected. Consequently, the conjecture of Wei and Zhu is proved as NC ₂(G) ≥ NC(G). Furthermore, we show that the lower bound is best possible and characterize all 2-connected graphs with NC ₂(G) ≥ n − δ(G) + 1 which are not [4, n]-pan-connected.     

Minimum degree conditions for -linked graphs

For a fixed multigraph H with vertices w₁,…,w_m, a graph G is H-linked if for every choice of vertices v₁,…,v_m in G, there exists a subdivision of H in G such that v_i is the branch vertex representing w_i (for all i). This generalizes the notions of k-linked, k-connected, and k-ordered graphs.Given a connected multigraph H with k edges and minimum degree at least two and n7.5k, we determine the least integer d such that every n-vertex simple graph with minimum degree at least d is H-linked. This value D(H,n) appears to equal the least integer d^′ such that every n-vertex graph with minimum degree at least d^′ is b(H)-connected, where b(H) is the maximum number of edges in a bipartite subgraph of H.     

Hyper- and reverse-Wiener indices of F-sums of graphs

The Wiener index W(G)=∑_{u,v}⊂V(G)d(u,v), the hyper-Wiener index and the reverse-Wiener index , where d(u,v) is the distance of two vertices u,v in G, d²(u,v)=d(u,v)², n=|V(G)| and D is the diameter of G. In [M. Eliasi, B. Taeri, Four new sums of graphs and their Wiener indices, Discrete Appl. Math. 157 (2009) 794-803], Eliasi and Taeri introduced the F-sums of two connected graphs. In this paper, we determine the hyper- and reverse-Wiener indices of the F-sum graphs and, subject to some condition, we present some exact expressions of the reverse-Wiener indices of the F-sum graphs.     

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.     

Spanning k-Trees of n-Connected Graphs

A tree is called a k-tree if the maximum degree is at most k. We prove the following theorem, by which a closure concept for spanning k-trees of n-connected graphs can be defined. Let k ≥ 2 and n ≥ 1 be integers, and let u and v be a pair of nonadjacent vertices of an n-connected graph G such that deg _G (u) + deg _G (v) ≥ |G| − 1 − (k − 2)n, where |G| denotes the order of G. Then G has a spanning k-tree if and only if G + uv has a spanning k-tree.