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Intersection Dimension and Maximum Degree
"... Abstract We show that the intersection dimension of graphs with respect to several hereditary graph classes can be bounded as a function of the maximum degree. As an interesting special case, we show that the circular dimension of a graph with maximum degree ∆ is at most O(∆ log ∆ log log ∆ ). We a ..."
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Abstract We show that the intersection dimension of graphs with respect to several hereditary graph classes can be bounded as a function of the maximum degree. As an interesting special case, we show that the circular dimension of a graph with maximum degree ∆ is at most O(∆ log ∆ log log ∆ ). We
Note Boxicity and maximum degree
, 2008
"... A ddimensional box is a Cartesian product of d closed intervals on the real line. The boxicity of a graph is the minimum dimension d such that it is representable as the intersection graph of ddimensional boxes. We give a short constructive proof that every graph with maximum degree D has boxicity ..."
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A ddimensional box is a Cartesian product of d closed intervals on the real line. The boxicity of a graph is the minimum dimension d such that it is representable as the intersection graph of ddimensional boxes. We give a short constructive proof that every graph with maximum degree D has
Boxicity and Maximum degree
"... An axisparallel d–dimensional box is a Cartesian product R1 × R2 × · · · × Rd where Ri (for 1 ≤ i ≤ d) is a closed interval of the form [ai, bi] on the real line. For a graph G, its boxicity box(G) is the minimum dimension d, such that G is representable as the intersection graph of (axis–paral ..."
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Cited by 9 (5 self)
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–parallel) boxes in d–dimensional space. The concept of boxicity finds applications in various areas such as ecology, operation research etc. We show that for any graph G with maximum degree ∆, box(G) ≤ 2 ∆ 2 + 2. That the bound does not depend on the number of vertices is a bit surprising considering the fact
Maximum degree in graphs of diameter 2
 Networks
, 1980
"... It is well known that there are at most four Moore graphs of diameter 2, i.e., graphs of diameter 2, maximum degree d, and d 2 + 1 vertices. The purpose of this paper is to prove that with the exception of C4, there are no graphs of diameter 2, of maximum degree d, and with d 2 vertices. ..."
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Cited by 15 (0 self)
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It is well known that there are at most four Moore graphs of diameter 2, i.e., graphs of diameter 2, maximum degree d, and d 2 + 1 vertices. The purpose of this paper is to prove that with the exception of C4, there are no graphs of diameter 2, of maximum degree d, and with d 2 vertices.
On maximum degree energy of a graph
 Int. J. Contemp. Math. Sci
, 2009
"... Abstract In this paper we introduce the concept of maximum degree matrix M (G) of a simple graph G and obtain a bound for eigenvalues of M (G). We also introduce maximum degree energy E M (G) of a graph G and obtain bounds for E M (G). We prove that the maximum degree energies of certain classes of ..."
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Cited by 3 (0 self)
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Abstract In this paper we introduce the concept of maximum degree matrix M (G) of a simple graph G and obtain a bound for eigenvalues of M (G). We also introduce maximum degree energy E M (G) of a graph G and obtain bounds for E M (G). We prove that the maximum degree energies of certain classes
On the Maximum Degree of Minimum Spanning Trees
 IN PROC. ACM SYMP. COMPUTATIONAL GEOMETRY, STONY
, 1994
"... Motivated by practical VLSI routing applications, we study the maximum vertex degree of a minimum spanning tree (MST). We prove that under the L p norm, the maximum vertex degree over all MSTs is equal to the Hadwiger number of the corresponding unit ball; we show an even tighter bound for MSTs whe ..."
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Cited by 11 (3 self)
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Motivated by practical VLSI routing applications, we study the maximum vertex degree of a minimum spanning tree (MST). We prove that under the L p norm, the maximum vertex degree over all MSTs is equal to the Hadwiger number of the corresponding unit ball; we show an even tighter bound for MSTs
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