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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 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...
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...
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...
We define some infinite subfamily of hexagonal grids with the oriented chromatic number 5. We present an algorithm for oriented colouring of some hexagonal planar oriented grids. The algorithm uses BFS spanning tree of a subgraph of the dual graph of the grid and a homomorphism to some tournament of order 6. In general the difference between the number of colours given by the algorithm and the oriented...
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.
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