scholarly journals The Energy Change of the Complete Multipartite Graph

2020 ◽  
Vol 36 (36) ◽  
pp. 309-317
Author(s):  
Haiying Shan ◽  
Changxiang He ◽  
Zhensheng Yu
Keyword(s):  
Linear Algebra ◽  
Singular Values ◽  
New Method ◽  
Energy Change ◽  
Natural Question ◽  
Complete Multipartite Graph ◽  
Multipartite Graph ◽  
Energy Of Graph ◽  
Complete Multipartite ◽  
Edge Addition

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. Akbari et al. [S. Akbari, E. Ghorbani, and M. Oboudi. Edge addition, singular values, and energy of graphs and matrices. {\em Linear Algebra Appl.}, 430:2192--2199, 2009.] proved that for a complete multipartite graph $K_{t_1 ,\ldots,t_k}$, if $t_i\geq 2 \ (i=1,\ldots,k)$, then deleting any edge will increase the energy. A natural question is how the energy changes when $\min\{t_1 ,\ldots,t_k\}=1$. In this paper, a new method to study the energy of graph is explored. As an application of this new method, the above natural question is answered and it is completely determined how the energy of a complete multipartite graph changes when one edge is removed.

scholarly journals A shorter proof of the distance energy of complete multipartite graphs

2017 ◽  
Vol 5 (1) ◽  
pp. 61-63 ◽  
Author(s):  
Wasin So
Keyword(s):  
Linear Algebra ◽  
Symmetric Matrix ◽  
Complete Multipartite Graph ◽  
Real Symmetric Matrix ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Multipartite Graph ◽  
Complete Multipartite

Abstract Caporossi, Chasser and Furtula in [Les Cahiers du GERAD (2009) G-2009-64] conjectured that the distance energy of a complete multipartite graph of order n with r ≥ 2 parts, each of size at least 2, is equal to 4(n − r). Stevanovic, Milosevic, Hic and Pokorny in [MATCH Commun. Math. Comput. Chem. 70 (2013), no. 1, 157-162.] proved the conjecture, and then Zhang in [Linear Algebra Appl. 450 (2014), 108-120.] gave another proof. We give a shorter proof of this conjecture using the interlacing inequalities of a positve semi-definite rank-1 perturbation to a real symmetric matrix.

scholarly journals The Size, Multipartite Ramsey Numbers for nK2 Versus Path–Path and Cycle

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 764
Author(s):  
Yaser Rowshan ◽  
Mostafa Gholami ◽  
Stanford Shateyi
Keyword(s):  
Positive Integer ◽  
Ramsey Number ◽  
Complete Multipartite Graph ◽  
Ramsey Numbers ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Monochromatic Copy

For given graphs G1,G2,…,Gn and any integer j, the size of the multipartite Ramsey number mj(G1,G2,…,Gn) is the smallest positive integer t such that any n-coloring of the edges of Kj×t contains a monochromatic copy of Gi in color i for some i, 1≤i≤n, where Kj×t denotes the complete multipartite graph having j classes with t vertices per each class. In this paper, we computed the size of the multipartite Ramsey numbers mj(K1,2,P4,nK2) for any j,n≥2 and mj(nK2,C7), for any j≤4 and n≥2.

Codes from multipartite graphs and minimal permutation decoding sets

2015 ◽  
Vol 07 (04) ◽  
pp. 1550060
Author(s):  
P. Seneviratne
Keyword(s):  
Automorphism Group ◽  
Error Correcting Code ◽  
Binary Codes ◽  
Hard Problem ◽  
Complete Multipartite Graph ◽  
Decoding Algorithm ◽  
Permutation Decoding ◽  
Multipartite Graphs ◽  
Multipartite Graph ◽  
Complete Multipartite

Permutation decoding method developed by MacWilliams and described in [Permutation decoding of systematic codes, Bell Syst. Tech. J. 43 (1964) 485–505] is a decoding technique that uses a subset of the automorphism group of the code called a PD-set. The complexity of the permutation decoding algorithm depends on the size of the PD-set and finding a minimal PD-set for an error correcting code is a hard problem. In this paper we examine binary codes from the complete-multipartite graph [Formula: see text] and find PD-sets for all values of [Formula: see text] and [Formula: see text]. Further we show that these PD-sets are minimal when [Formula: see text] is odd and [Formula: see text].

scholarly journals Cyclic cycle systems of the complete multipartite graph

2019 ◽  
Vol 28 (3) ◽  
pp. 224-260
Author(s):  
Andrea Burgess ◽  
Francesca Merola ◽  
Tommaso Traetta
Keyword(s):  
Complete Multipartite Graph ◽  
Cycle Systems ◽  
Multipartite Graph ◽  
Complete Multipartite

scholarly journals On claw-decomposition of a complete multipartite graph

1978 ◽  
Vol 8 (1) ◽  
pp. 207-210 ◽  
Author(s):  
Kazuhiko Ushio ◽  
Shinsei Tazawa ◽  
Sumiyasu Yamamoto
Keyword(s):  
Complete Multipartite Graph ◽  
Multipartite Graph ◽  
Complete Multipartite
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