We prove the conjecture of Albertson stating that every planar graph can be 5-list-colored, even if it contains precolored vertices, as long as they are sufficiently far apart from each other. In order to prove this claim, we also give bounds on the sizes of graphs critical with respect to 5-list coloring. In particular, if G is a planar graph, H is a connected subgraph of G and L is an assignm...

A class of graph invariants referred to today as topological indices are inefficient progressively acknowledged by scientific experts and others be integral assets in the depiction structural phenomena. The structure an interconnection network can represented a graph. In network, vertices represent processor nodes edges links between nodes. Graph play vital feature theory distinguish properties...

for a simple digraph $g$ of order $n$ with vertex set${v_1,v_2,ldots, v_n}$, let $d_i^+$ and $d_i^-$ denote theout-degree and in-degree of a vertex $v_i$ in $g$, respectively. let$d^+(g)=diag(d_1^+,d_2^+,ldots,d_n^+)$ and$d^-(g)=diag(d_1^-,d_2^-,ldots,d_n^-)$. in this paper we introduce$widetilde{sl}(g)=widetilde{d}(g)-s(g)$ to be a new kind of skewlaplacian matrix of $g$, where $widetilde{d}(g...

abstract intensive and inefficient use of energy leads to degradation of natural resources and environmental problems. this study evaluated the relationship between the energy of consumed inputs and yield of tomato, cucumber and melon using the cobb–douglas production function. the data were collected from 90 farmers in firoozabad using a questionnaire. the farmers were chosen by simple random ...

The study of topological indices in graph theory is one the more important topics, as scientific development that occurred previous century had an impact by linking it to many chemical and physical properties such boiling point melting point. So, our interest this paper “generalized indices’ network” for some graphs have somewhat strange structure, so called cog-graphs special “molecular networ...