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.     

Line graphs of complete multipartite graphs have small cycle double covers

It has been shown by MacGillivray and Seyffarth (Austral. J. Combin. 24 (2001) 91) that bridgeless line graphs of complete graphs, complete bipartite graphs, and planar graphs have small cycle double covers. In this paper, we extend the result for complete bipartite graphs, and show that the line graph of any complete multipartite graph (other than K_1,2) has a small cycle double cover.     

完全多部图的无符号Laplacian特征多项式(英文)

For a simple graph G,let matrix Q（G）=D（G） + A（G） be it’s signless Laplacian matrix and Q G （λ）=det（λI Q） it’s signless Laplacian characteristic polynomial,where D（G） denotes the diagonal matrix of vertex degrees of G,A（G） denotes its adjacency matrix of G.If all eigenvalues of Q G （λ） are integral,then the graph G is called Q-integral.In this paper,we obtain that the signless Laplacian characteristic polynomials of the complete multi-partite graphs G=K（n₁,n₂,···,n_t）.We prove that the complete t-partite graphs K（n,n,···,n）t are Q-integral and give a necessary and sufficient condition for the complete multipartite graphs K（m,···,m）s（n,···,n）t to be Q-integral.We also obtain that the signless Laplacian characteristic polynomials of the complete multipartite graphs K（m,···,m,）s1（n,···,n,）s2（l,···,l）s3.     

Locally s‐distance transitive graphs

We give a unified approach to analyzing, for each positive integer s, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally s‐arc transitive graphs of diameter at least s. A graph is in the class if it is connected and if, for each vertex v, the subgroup of automorphisms fixing v acts transitively on the set of vertices at distance i from v, for each i from 1 to s. We prove that this class is closed under forming normal quotients. Several graphs in the class are designated as degenerate, and a nondegenerate graph in the class is called basic if all its nontrivial normal quotients are degenerate. We prove that, for s≥2, a nondegenerate, nonbasic graph in the class is either a complete multipartite graph or a normal cover of a basic graph. We prove further that, apart from the complete bipartite graphs, each basic graph admits a faithful quasiprimitive action on each of its (1 or 2) vertex‐orbits or a biquasiprimitive action. These results invite detailed additional analysis of the basic graphs using the theory of quasiprimitive permutation groups. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:176‐197, 2012     

On normal 2-geodesic transitive Cayley graphs

We investigate connected normal 2-geodesic transitive Cayley graphs Cay(T,S). We first prove that if Cay(T,S) is neither cyclic nor K_4[2], then 〈a〉?{1}??S for all a∈S. Next, as an application, we give a reduction theorem proving that each graph in this family which is neither a complete multipartite graph nor a bipartite 2-arc transitive graph, has a normal quotient that is either a complete graph or a Cayley graph in the family for a characteristically simple group. Finally we classify complete multipartite graphs in the family.     

On the asymptotic behavior of graphs determined by their generalized spectra

In Wang and Xu (2006) [15] and [16] the authors introduced a family of graphs H_n and gave some methods for finding graphs among this family that are determined by their generalized spectra. This paper is a continuation of our previous work. We further show that almost all graphs in H_n are determined by their generalized spectra. This gives some evidences for the conjecture that almost all graphs are determined by their generalized spectra.     

Complete multipartite decompositions of complete graphs and complete <Emphasis Type="Italic">n</Emphasis>-partite graphs

In this paper, a new concept of an optimal complete multipartite decomposition of type 1 (type 2) of a complete n-partite graph Q _n is proposed and another new concept of a normal complete multipartite decomposition of K _n is introduced. It is showed that an optimal complete multipartite decomposition of type 1 of K _n is a normal complete multipartite decomposition. As for any complete multipartite decomposition of K _n , there is a derived complete multipartite decomposition for Q _n . It is also showed that any optimal complete multipartite decomposition of type 1 of Q _n is a derived decomposition of an optimal complete multipartite decomposition of type 1 of K _n . Besides, some structural properties of an optimal complete multipartite decomposition of type 1 of K _n are given. Supported by the National Natural Science Foundation of China (10271110).     

Distance-hereditary graphs

Distance-hereditary graphs (sensu Howorka) are connected graphs in which all induced paths are isometric. Examples of such graphs are provided by complete multipartite graphs and ptolemaic graphs. Every finite distance-hereditary graph is obtained from K₁ by iterating the following two operations: adding pendant vertices and splitting vertices. Moreover, distance-hereditary graphs are characterized in terms of the distance function d, or via forbidden isometric subgraphs.     

A new arithmetic criterion for graphs being determined by their generalized Q-spectrum

“Which graphs are determined by their spectrum (DS for short)?” is a fundamental question in spectral graph theory. It is generally very hard to show a given graph to be DS and few results about DS graphs are known in literature. In this paper, we consider the above problem in the context of the generalized $Q$-spectrum. A graph $G$ is said to be determined by the generalized $Q$-spectrum (DGQS for short) if, for any graph $H$, $H$ and $G$ have the same $Q$-spectrum and so do their complements imply that $H$ is isomorphic to $G$. We give a simple arithmetic condition for a graph being DGQS. More precisely, let $G$ be a graph with adjacency matrix $A$ and degree diagonal matrix $D$. Let $Q=A+D$ be the signless Laplacian matrix of $G$, and $W_Q\left(G\right)=[e,Qe,\dots ,Q^{n-1}e]$ ($e$ is the all-ones vector) be the $Q$-walk matrix. We show that if $\frac{det{W}_Q\left(G\right)}{2^{\lfloor \frac{3n-2}{2}\rfloor }}$ (which is always an integer) is odd and square-free, then $G$ is DGQS.     

6-Transitive graphs

A connected graph is n-transitive if, whenever two n-tuples are isometric, there is an automorphism mapping the first to the second. It is shown that a 6-transitive graph is complete multipartite, or complete bipartite with a matching deleted, or a cycle, or one of three special graphs on 9, 12 and 20 vertices. These graphs are n-transitive for all n; but there are graphs (the smallest on 56 vertices) which are 5- but not 6-transitive.     

Clique-critical graphs: Maximum size and recognition

The clique graph of G, K(G), is the intersection graph of the family of cliques (maximal complete sets) of G. Clique-critical graphs were defined as those whose clique graph changes whenever a vertex is removed. We prove that if G has m edges then any clique-critical graph in K^-1(G) has at most 2m vertices, which solves a question posed by Escalante and Toft [On clique-critical graphs, J. Combin. Theory B 17 (1974) 170-182]. The proof is based on a restatement of their characterization of clique-critical graphs. Moreover, the bound is sharp. We also show that the problem of recognizing clique-critical graphs is NP-complete.