On the <i>r</i>-dynamic coloring of some fan graph families

Ganfornina, Raúl Manuel Falcón; Venkatachalam, M.; Gowri, S.; Nandini, G.

doi:10.2478/auom-2021-0039

Cited by 2 publications

(2 citation statements)

References 28 publications

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“…The aim of this paper is to obtain explicitly the QE constant of a fan graph K 1 + P n , that is the graph join of the singleton graph K 1 and the path P n with n ≥ 1. The fan graphs form a basic family of graphs and have been studied in various contexts, for example, vertex coloring [7], edge coloring [8], chromatic polynomials [16], inversion of the distance matrices [9], graph spectrum [14] and so forth. Although a fan graph has a relatively simple structure at a first glance, no reasonable formula for QEC(K 1 + P n ) has been obtained.…”

Section: Introductionmentioning

confidence: 99%

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On quadratic embedding constants of star product graphs

Młotkowski

Obata

2020

Hokkaido Math. J.

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We derive a general formula for the quadratic embedding constant of a graph join Km + G, where Km is the empty graph on m ≥ 1 vertices and G is an arbitrary graph. Applying our formula to a fan graph K 1 + P n , where K 1 = K1 is the singleton graph and P n is the path on n ≥ 1 vertices, we show that QEC(K 1 + P n ) = − αn − 2, where αn is the minimal zero of a new polynomial Φ n (x) related to Chebyshev polynomials of the second kind. Moreover, for an even n we have αn = min ev(A n ), where the right-hand side is the An minimal eigenvalue of the adjacency matrix A n of P n . For an odd n we show that min ev(A n+1 ) ≤ αn < min ev(A n ).

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Section: Introductionmentioning

confidence: 99%

“…The formula is stated in Theorem 3. 7 and some examples are given for illustrating how our formula works. In Section 4 we prove the main result on QEC(K 1 + P n ), see Theorem 4.8.…”

Section: Introductionmentioning

confidence: 99%

On quadratic embedding constants of star product graphs

Młotkowski

Obata

2020

Hokkaido Math. J.

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Optimal secret share distribution in degree splitting communication networks

Falcón,

Aparna,

Mohanapriya

2023

NHM

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<abstract><p>Dynamic coloring has recently emerged as a valuable tool to optimize cryptographic protocols based on secret sharing, which enforce data security in communication networks and have significant importance in both online storage and cloud computing. This type of graph labeling enables the dealer to distribute secret shares among the nodes of a communication network so that everybody can recover the secret after a minimum number of rounds of communication. This paper delves into this topic by dealing with the dynamic coloring problem for degree splitting graphs. The topological structure of the latter enables the dealer to avoid dishonesty by adding control nodes that supervise all those participants with a similar influence in the network. More precisely, we solve the dynamic coloring problem for degree splitting graphs of any regular graph. The irregular case is partially solved by establishing a lower bound for the corresponding dynamic chromatic number. As illustrative examples, we solve the dynamic coloring problem for the degree splitting graphs of cycles, cocktail, book, comb, fan, jellyfish, windmill and barbell graphs.</p></abstract>

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On the r-dynamic coloring of some fan graph families

Cited by 2 publications

References 28 publications

On quadratic embedding constants of star product graphs

On quadratic embedding constants of star product graphs

Optimal secret share distribution in degree splitting communication networks

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