Products of distance degree regular and distance degree injective graphs

Abstract

The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &

Distance Degree Regular Graphs and Distance Degree Injective Graphs: An Overview

“…On the other extreme, construction of new DDI graphs from the smaller order/diameter is also interesting and is looked into by Huilgol et al [23]. In this paper many results are discussed for different products and the behavior of DDI graphs with respect to these products.…”

“…Proposition 21 (see [23]). There exists a DDR graph of diameter ≥ 3 of order = (2 + 1) and = 2 , where = 1, 2, 3, .…”

“…Theorem 26 (see [23]). Normal product 1 ⊕ 2 of two graphs 1 and 2 is DDR if and only if both 1 and 2 are DDR graphs.…”

“…In [23], Huilgol et al have constructed higher order DDR graphs by considering the simplest of the products, namely, the cartesian and normal product. The cartesian product of two graphs and , denoted by ◻ , is a graph with vertex set ( ◻ ) = ( ) × ( ); that is, the set {( , ℎ)/ ∈ ( ), ℎ ∈ ( )}.…”

“…And the normal product of two graphs and , denoted by ⊕ , is a graph with vertex set ( ⊕ ) = ( ) × ( ); that is, the set {( , ℎ)/ ∈ ( ), ℎ ∈ ( )}, and an edge [( 1 , ℎ 1 ), ( 2 , ℎ 2 )] exists whenever any of the following conditions hold good: (i) [ 1 , 2 ] ∈ ( ) and ℎ 1 = ℎ 2 , (ii) Theorem 24 (see [23]). Cartesian product of two graphs 1 and 2 is a DDR graph if and only if both 1 and 2 are DDR graphs.…”