Let G = (V,E) be a simple connected graph. The eccentric-distance sum of G is defined as ξ(G) = ∑ {u,v}⊆V (G) [e(u) + e(v)]d(u, v), where e(u) is the eccentricity of the vertex u in G and d(u, v) is the distance between u and v. In this paper, we establish formulae to calculate the eccentric-distance sum for some graphs, namely wheel, star, broom, lollipop, double star, friendship, multi-star g...

A chord of a path $P$ is an edge joining two non-adjacent vertices of $P$. A path  $P$ is called a monophonic path if it is a chordless path. A longest $x-y$ monophonic path is called an $x-y$ detour monophonic path. A  detour monophonic graphoidal cover of a graph $G$ is a collection $psi_{dm}$ of detour monophonic paths in $G$ such that every vertex of $G$ is an internal vertex  of at most on...

Given a configuration of pebbles on the vertices of a graph, a pebbling move is defined by removing two pebbles from some vertex and placing one pebble on an adjacent vertex. The cover pebbling number of a graph, γ(G), is the smallest number of pebbles such that through a sequence of pebbling moves, a pebble can eventually be placed on every vertex simultaneously, no matter how the pebbles are ...

This study aims to explain the phenomenon of motion in hamsters moving a rolling wheel. Motion is one materials taught at 7th-grade junior high school. The can be used learning with practicum method, namely animal motion. Hamster (Cricetinae) analyzed using video tracker analysis application javascript program. direct experiment hamster four reference points, forelegs, head, middle body, and hi...

Let $W_t$ denote the wheel on $t+1$ vertices. We prove that for every integer $t \geq 3$ there is a constant $c=c(t)$ such that for every integer $k\geq 1$ and every graph $G$, either $G$ has $k$ vertex-disjoint subgraphs each containing $W_t$ as minor, or there is a subset $X$ of at most $c k \log k$ vertices such that $G-X$ has no $W_t$ minor. This is best possible, up to the value of $c$. We...

For a median graph G and a vertex v of G that is not a cut-vertex we show that G − v is a median graph precisely when v is not the center of a bipartite wheel, which is in turn equivalent with the existence of a certain edge elimination scheme for edges incident with v. This implies a characterization of vertex-critical (respectively, vertex-complete) median graphs, which are median graphs whos...

For a connected graph G, a set of vertices W in G is called a Steiner dominating set if W is both a Steiner set and a dominating set. The minimum cardinality of a Steiner dominating set of G is its Steiner domination numberand is denoted by ) (G s  . In this paper, it is studied that how the Steiner domination number is affected by adding a single edge to paths, complete graphs, cycles, star a...

It was conjectured by Paul Erdős that if G is a graph with chromatic number at least k, then the diagonal Ramsey number r(G) ≥ r(Kk). That is, the complete graph Kk has the smallest diagonal Ramsey number among the graphs of chromatic number k. This conjecture is shown to be false for k = 4 by verifying that r(W6) = 17, where W6 is the wheel with 6 vertices, since it is well known that r(K4) = ...

OF THE DISSERTATION A Combinatorial Comparison of Elliptic Curves and Critical Groups of Graphs by Gregg Joseph Musiker Doctor of Philosophy in Mathematics University of California San Diego, 2007 Professor Adriano Garsia, Chair In this thesis, we explore elliptic curves from a combinatorial viewpoint. Given an elliptic curve E, we study here Nk = #E(Fqk), the number of points of E over the fin...

In the previous paper, it was shown that the graph U. ÷ 1 obtained from the wheel W n ÷ 1 by deleting a spoke is uniquely determined by its chromatic polynomial if n >i 3 is odd. In this paper, we show that the result is also true for even n >~ 4 except when n = 6 in which case, the graph W given in the paper is the only graph having the same chromatic polynomial as that of U 7. The relevant to...