Remarks on D-integral complete multipartite graphs

A graph is called distance integral (or D-integral) if all eigenvalues of its distance matrix are integers. In their study of D-integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on D-integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs \({K_{{p_1},{p_2},{p_3}}}\) with p₁ < p₂ < p₃, and \({K_{{p_1},{p_2},{p_3},{p_4}}}\) with p₁ < p₂ < p₃ < p₄, as well as the infinite classes of distance integral complete multipartite graphs \({K_{{a_1}{p_1},{a_2}{p_2},...,{a_s}{p_s}}}\) with s = 5, 6.     

On the seidel integral complete multipartite graphs

For a simple undirected graph G, denote by A(G) the (0,1)-adjacency matrix of G. Let thematrix S(G) = J-I-2A(G) be its Seidel matrix, and let S _G (??) = det(??I-S(G)) be its Seidel characteristic polynomial, where I is an identity matrix and J is a square matrix all of whose entries are equal to 1. If all eigenvalues of S _G (??) are integral, then the graph G is called S-integral. In this paper, our main goal is to investigate the eigenvalues of S _G (??) for the complete multipartite graphs G = $G = K_{n_1 ,n_2 ,...n_t } $ . A necessary and sufficient condition for the complete tripartite graphs K _m,n,t and the complete multipartite graphs to be S-integral is given, respectively.     

Connected Q-integral graphs with maximum edge-degree less than or equal to 8

Graphs with integral signless Laplacian spectrum are called Q-integral graphs. The number of adjacent edges to an edge is defined as the edge-degree of that edge. The Q-spectral radius of a graph is the largest eigenvalue of its signless Laplacian. In 2019, Park and Sano [16] studied connected Q-integral graphs with the maximum edge-degree at most six. In this article, we extend their result and study the connected Q-integral graphs with maximum edge-degree less than or equal to eight. Further, we give an upper bound and a lower bound for the maximum edge-degree of a connected Q-integral graph with respect to its Q-spectral radius. As a corollary, we show that the Q-spectral radius of the connected edge-non-regular Q-integral graph with maximum edge-degree five is six, which we anticipate to be a key for solving the unsolved problem of characterizing such graphs.     

On the spectral characterization of the union of complete multipartite graph and some isolated vertices

A graph is said to be determined by its adjacency spectrum (DS for short) if there is no other non-isomorphic graph with the same spectrum. In this paper, we focus our attention on the spectral characterization of the union of complete multipartite graph and some isolated vertices, and all its cospectral graphs are obtained. By the results, some complete multipartite graphs determined by their adjacency spectrum are also given. This extends several previous results of spectral characterization related to the complete multipartite graphs.     

Line-graphs of cubic graphs are normal

A graph is called normal if its vertex set can be covered by cliques Q₁,Q₂,…,Q_k and also by stable sets S₁,S₂,…,S_l, such that S_i∩Q_j≠∅ for every i,j. This notion is due to Körner, who introduced the class of normal graphs as an extension of the class of perfect graphs. Normality has also relevance in information theory. Here we prove, that the line graphs of cubic graphs are normal.     

Polar cographs

Polar graphs are a natural extension of some classes of graphs like bipartite graphs, split graphs and complements of bipartite graphs. A graph is (s,k)-polar if there exists a partition A,B of its vertex set such that A induces a complete s-partite graph (i.e., a collection of at most s disjoint stable sets with complete links between all sets) and B a disjoint union of at most k cliques (i.e., the complement of a complete k-partite graph).Recognizing a polar graph is known to be NP-complete. These graphs have not been extensively studied and no good characterization is known. Here we consider the class of polar graphs which are also cographs (graphs without induced path on four vertices). We provide a characterization in terms of forbidden subgraphs. Besides, we give an algorithm in time O(n) for finding a largest induced polar subgraph in cographs; this also serves as a polar cograph recognition algorithm. We examine also the monopolar cographs which are the (s,k)-polar cographs where min(s,k)?1. A characterization of these graphs by forbidden subgraphs is given. Some open questions related to polarity are discussed.     

On conservative and supermagic graphs

A graph is called supermagic if it admits a labelling of the edges by pairwise different consecutive positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. A graph G is called conservative if it admits an orientation and a labelling of the edges by integers {1,…,|E(G)|} such that at each vertex the sum of the labels on the incoming edges is equal to the sum of the labels on the outgoing edges. In this paper we deal with conservative graphs and their connection with the supermagic graphs. We introduce a new method to construct supermagic graphs using conservative graphs. Inter alia we show that the union of some circulant graphs and regular complete multipartite graphs are supermagic.     

Q-integral graphs with edge-degrees at most five

We consider the problem of determining the Q-integral graphs, i.e. the graphs with integral signless Laplacian spectrum. We find all such graphs with maximum edge-degree 4, and obtain only partial results for the next natural case, with maximum edge-degree 5.     

The Inducibility of Graphs on Four Vertices

We consider the problem of determining the maximum induced density of a graph H in any graph on n vertices. The limit of this density as n tends to infinity is called the inducibility of H. The exact value of this quantity is known only for a handful of small graphs and a specific set of complete multipartite graphs. Answering questions of Brown–Sidorenko and Exoo, we determine the inducibility of K_{1, 1, 2} and the paw graph. The proof is obtained using semidefinite programming techniques based on a modern language of extremal graph theory, which we describe in full detail in an accessible setting.     

On choosability of some complete multipartite graphs and Ohba's conjecture

A graph G is said to be chromatic-choosable if ch(G)=χ(G). Ohba has conjectured that every graph G with 2χ(G)+1 or fewer vertices is chromatic-choosable. It is clear that Ohba's conjecture is true if and only if it is true for complete multipartite graphs. But for complete multipartite graphs, the graphs for which Ohba's conjecture has been verified are nothing more than K_{3*2,2*(k-3),1}, K_3,2*(k-1), and K_{s+3,2*(k-s-1),1*s}. These results have been obtained indirectly from the investigation about complete multipartite graphs by Gravier and Maffray and by Enomoto et al. In this paper we show that Ohba's conjecture is true for complete multipartite graphs K_{4,3,2*(k-4),1*2} and K_{5,3,2*(k-5),1*3}. By the way, we give some discussions about a result of Enomoto et al.