A new method for counting chromatic coefficients

In this paper, Proper-Broken-Cycle Formula is presented. The explicit expression in terms of induced subgraphs for the sixth coefficient of chromatic polynomial of a graph is presented. Also a new proof of Farrell's theorems is given

The Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths

The Ramsey number $R(G, H)$ for a pair of graphs $G$ and $H$ is defined as the smallest integer $n$ such that, for any graph $F$ on $n$ vertices, either $F$ contains $G$ or $\overline{F}$ contains $H$ as a subgraph, where $\overline{F}$ denotes the complement of $F$. We study Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths and determine these numbers for some cases. We extend many known results studied in [5, 14, 18, 19, 20]. In particular we count the numbers $R(K_1+L_n, P_m)$ and $R(K_1+L_n, C_m)$ for some integers $m$, $n$, where $L_n$ is a linear forest of order $n$ with at least one edge

Statuses and double branch weights of quadrangular outerplanar graphs

In this paper we study some distance properties of outerplanar graphs with the Hamiltonian cycle whose all bounded faces are cycles isomorphic to the cycle C4. We call this family of graphs quadrangular outerplanar graphs. We give the lower and upper bound on the double branch weight and the status for this graphs. At the end of this paper we show some relations between median and double centroid in quadrangular outerplanar graphs

Eccentric distance sum index for some classes of connected graphs

In this paper we show some properties of the eccentric distance sum index which is defined as follows $\xi^{d}(G)=\sum_{v \in V(G)}D(v) \varepsilon(v)$. This index is widely used by chemists and biologists in their researches. We present a lower bound of this index for a new class of graphs

On the adjacent eccentric distance sum of graphs

In this paper we show bounds for the adjacent eccentric distance sum of graphs in terms of Wiener index, maximum degree and minimum degree. We extend some earlier results of Hua and Yu [Bounds for the Adjacent Eccentric Distance Sum, International Mathematical Forum, Vol. 7 (2002) no. 26, 1289–1294]. The adjacent eccentric distance sum index of the graph $G$ is defined as$$\xi ^{sv} (G)= \sum_{v\in V(G)}{\frac{\varepsilon (v) D(v)}{deg(v)}},$$where $\varepsilon(v)$ is the eccentricity of the vertex $v$, $deg(v)$ is the degree of the vertex $v$ and$$D(v)=\sum_{u\in V(G)}{d(u,v)}$$is the sum of all distances from the vertex $v$

The density Turan problem for 3-uniform linear hypertrees. An efficient testing algorithm

Let $\mathcal{T}=(V,\mathcal{E})$ be a  3-uniform linear hypertree. We consider a blow-up hypergraph $\mathcal{B}[\mathcal{T}]$. We are interested in the following problem. We have to decide whether there exists a blow-up hypergraph $\mathcal{B}[\mathcal{T}]$ of the hypertree $\mathcal{T}$, with hyperedge densities satisfying some conditions, such that the hypertree $\mathcal{T}$ does not appear in a blow-up hypergraph as a transversal. We present an efficient algorithm to decide whether a given set of hyperedge densities ensures the existence of a 3-uniform linear hypertree $\mathcal{T}$ in a blow-up hypergraph $\mathcal{B}[\mathcal{T}]$

The Turán number of the graph 3P43P_4

Let $ex(n, G)$ denote the maximum number of edges in a graph on $n$ vertices which does not contain $G$ as a subgraph. Let $P_i$ denote a path consisting of $i$ vertices and let $mP_i$ denote $m$ disjoint copies of $P_i$. In this paper we count $ex(n, 3P_4)$

A self-stabilizing algorithm for detecting fundamental cycles in a graph with DFS spanning tree given

This paper presents a linear time self-stabilizing algorithm for detecting the set offundamental cycles on an undirected connected graph modelling asynchronous distributed system.The previous known algorithm has O(n^2) time complexity, whereas we prove that this one stabilizesafter O(n) moves. The distributed adversarial scheduler is considered. Both algorithms assume thatthe depth-search spanning tree of the graph is given. The output is given in a distributed manner asa state of variables in the nodes

A self–stabilizing algorithm for finding weighted centroid in trees

In this paper we present some modification of the Blair and Manne algorithm for finding the center of a tree network in the distributed, self-stabilizing environment. Their algorithm finds n/2 -separator of a tree. Our algorithm finds weighted centroid, which is direct generalization of the former one for tree networks with positive weights on nodes. Time complexity of both algorithms is O(n2), where n is the number of nodes in the network

Closing the gap on path-kipas Ramsey numbers

Given two graphs G1 and G2, the Ramsey number R(G1,G2) is the smallest integer N such that, for any graph G of order N, either G1 is a subgraph of G, or G2 is a subgraph of the complement of G. Let Pn denote a path of order n and K^m a kipas of order m + 1, i.e., the graph obtained from a Pm by adding one new vertex v and edges from v to all vertices of the Pm. We close the gap in existing knowledge on exact values of the Ramsey numbers R(Pn,K^m) by determining the exact values for the remaining open cases