scholarly journals Unimodality of the independence polynomials of some composite graphs

Filomat ◽  
2017 ◽  
Vol 31 (3) ◽  
pp. 629-637 ◽  
Author(s):  
Bao-Xuan Zhu ◽  
Qinglin Lu
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Lexicographic Product ◽  
Complete Multipartite Graph ◽  
Real Zeros ◽  
Independence Polynomial ◽  
Independence Polynomials ◽  
Multipartite Graph ◽  
Complete Multipartite

Let I(G;x) denote the independence polynomial of a graph G. In this paper we study the unimodality properties of I(G; x) for some composite graphs G. Given two graphs G1 and G2, let G1[G2] denote the lexicographic product of G1 and G2. Assume I(G1; x) = P i_0 aixi and I(G2; x) = P i_0 bixi, where I(G2; x) is log-concave. Then we prove (i) if I(G1; x) is logconcave and (a2i ??ai??1ai+1)b21 _ aiai??1b2 for all 1 _ i _ _(G1), then I(G1[G2]; x) is log-concave; (ii) if ai??1 _ b1ai for 1 _ i _ _(G1), then I(G1[G2]; x) is unimodal. In particular, if ai is increasing in i, then I(G1[G2]; x) is unimodal. We also give two su_cient conditions when the independence polynomial of a complete multipartite graph is unimodal or log-concave. Finally, for every odd positive integer _ > 3, we find a connected graph G not a tree, such that _(G) = _, and I(G; x) is symmetric and has only real zeros. This answers a problem of Mandrescu and Miric?a.

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.

scholarly journals On-Line List Colouring of Complete Multipartite Graphs

10.37236/2050 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Seog-Jin Kim ◽  
Young Soo Kwon ◽  
Daphne Der-Fen Liu ◽  
Xuding Zhu
Keyword(s):  
Positive Integer ◽  
Complete Multipartite Graph ◽  
Minimal Function ◽  
Line List ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
On Line ◽  
Multipartite Graph ◽  
Complete Multipartite

The Ohba Conjecture says that every graph $G$ with $|V(G)| \le 2 \chi(G)+1$ is chromatic choosable. This paper studies an on-line version of Ohba Conjecture. We prove that unlike the off-line case, for $k \ge 3$, the complete multipartite graph $K_{2\star (k-1), 3}$ is not on-line chromatic-choosable. Based on this result, the on-line version of Ohba Conjecture is modified as follows: Every graph $G$ with $|V(G)| \le 2 \chi(G)$ is on-line chromatic choosable. We present an explicit strategy to show  that for any positive integer $k$, the graph $K_{2\star k}$ is on-line chromatic-choosable.  We then present a minimal function $g$ for which the graph $K_{2 \star (k-1), 3}$ is on-line $g$-choosable.

scholarly journals Regular graphs with regular neighborhoods

1992 ◽  
Vol 34 (2) ◽  
pp. 215-218
Author(s):  
Ľubomír Šoltés
Keyword(s):  
Connected Graph ◽  
Regular Graphs ◽  
Complete Multipartite Graph ◽  
Multipartite Graph ◽  
Complete Multipartite

The existence of r-regular graphs such that each edge lies in exactly t triangles, for given integers t < r, is studied. If t is sufficiently close to r then each such connected graph has to be the complete multipartite graph. Relations to graphs with isomorphic neighborhoods are also considered.

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 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 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

Unimodality of independence polynomials of rooted products of graphs

2019 ◽  
Vol 150 (5) ◽  
pp. 2573-2585
Author(s):  
Bao-Xuan Zhu ◽  
Qingxiu Wang
Keyword(s):  
Real Zeros ◽  
Independence Polynomial ◽  
Independence Polynomials

AbstractIn 1987, Alavi, Malde, Schwenk and Erdős conjectured that the independence polynomial of any tree is unimodal. Although it attracts many researchers' attention, it is still open. Motivated by this conjecture, in this paper, we prove that rooted products of some graphs preserve real rootedness of independence polynomials. As application, we not only give a unified proof for some known results, but also we can apply them to generate infinite kinds of trees whose independence polynomials have only real zeros. Thus their independence polynomials are unimodal.

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