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Since = if and only if the graph is bipartite, we will refer to the graphs that satisfy this alternative definition but not the first definition bipartite Ramanujan graphs. If G {\displaystyle G} is a Ramanujan graph, then G × K 2 {\displaystyle G\times K_{2}} is a bipartite Ramanujan graph, so the existence of Ramanujan graphs is stronger.
In the mathematical field of graph theory, the Chvátal graph is an undirected graph with 12 vertices and 24 edges, discovered by Václav Chvátal in 1970. It is the smallest graph that is triangle-free, 4-regular, and 4-chromatic.
Proof without words that a hypercube graph is non-planar using Kuratowski's or Wagner's theorems and finding either K 5 (top) or K 3,3 (bottom) subgraphs. The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of forbidden graphs, now known as Kuratowski's theorem:
The ANSI alphabetic state code is the same as the USPS state code except for U.S. Minor Outlying Islands, which have an ANSI code "UM" but no USPS code—and U.S. Military Mail locations, which have USPS codes ("AA", "AE", "AP") but no ANSI code.
James David "JD" Vance [a] (né Bowman; formerly Hamel; [b] born August 2, 1984) is an American politician, author, and Marine veteran who has served since 2023 as the junior United States senator from Ohio.
Stock market prices are often depicted with an open-high-low-close chart with a traditional bar chart of volume at the bottom. Candlestick charts are another type of bar chart used to describe price movements of an equity over time. A Kagi chart is a time-independent stock tracking chart that attempts to minimise noise.
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When the direction of a Euclidean vector is represented by an angle , this is the angle determined by the free vector (starting at the origin) and the positive -unit vector. The same concept may also be applied to lines in a Euclidean space, where the angle is that determined by a parallel to the given line through the origin and the positive x ...