Deza graphs with parameters (v,k,k-2,a)
Vladislav Kabanov, Leonid Shalaginov

TL;DR
This paper characterizes Deza graphs with parameters where the difference between the degree and the larger common neighbor count is two, expanding understanding beyond the previously studied case where this difference was one.
Contribution
It provides a complete characterization of Deza graphs with parameters satisfying $k-b=2$, extending prior work on the case $k-b=1$.
Findings
Deza graphs with $k-b=2$ are fully characterized.
The paper generalizes the classification of Deza graphs.
New structural properties of these graphs are identified.
Abstract
A Deza graph with parameters is a -regular graph on vertices in which the number of common neighbors of two distinct vertices takes two values or () and both cases exist. In the previous papers Deza graphs with parameters where were characterized. In the paper we characterise Deza graphs with .
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Deza graphs with parameters 111
Both authors are partially supported by RFBR according to the research project 17-51-560008.
Vladislav V. Kabanov
Leonid Shalaginov
Krasovskii Institute of Mathematics and Mechanics, S. Kovalevskaja st. 16, Yekaterinburg, 620990, Russia
Chelyabinsk State University, Brat’ev Kashirinyh st. 129, Chelyabinsk, 454021, Russia
Abstract
A Deza graph with parameters is a -regular graph on vertices in which the number of common neighbors of two distinct vertices takes two values or () and both cases exist. In the previous papers [9, 6] Deza graphs with parameters where were characterized. In this paper we characterise Deza graphs with .
keywords:
Deza graph; divisible design graph; strongly regular graph.
MSC:
[2010] 05C75, 05B30, 05E30
††journal: Journal of Combinatorial Designs
1 Introduction
The graphs studied in this paper are finite undirected graphs without loops and multiple edges. A Deza graph with parameters is a -regular graph on vertices in which the number of common neighbors of two distinct vertices takes two values or () and both cases exist.
The concept of a Deza graph was introduced by M. Erickson, S. Fernando, W. Haemers, D. Hardy, and J. Hemmeter in [5]. It was influenced by the paper of A. Deza and M. Deza [4].
In comparison with a strongly regular graph, a Deza graph (in case ) can have diameter more than 2. If a Deza graph has diameter 2 and is not strongly regular, then it is called a strictly Deza graph. If is a strongly regular graph, then the quadruple of parameters is used, where is equal to the number of common neighbors of every two adjacent vertices of and is equal to the number of common neighbors of every two distinct non-adjacent vertices of . Thus, the notion of Deza graphs is a generalization of the notion of strongly regular graphs in such a way that the number of common neighbors of any pair of distinct vertices in a Deza graph does not depend on the adjacency.
The authors of [5] developed a basic theory of strictly Deza graphs and introduced a few constructions of such graphs. They also found all strictly Deza graphs with the number of vertices at most 13. S. Goryainov and L. Shalaginov in [7] found all strictly Deza graphs which have the number of vertices equals to 14, 15, or 16. Deza graphs can have applications in several fields of discrete mathematics especially in design theory finite geometries, and connected point-block incidence structures.
A connected graph is called a -graph if any two distinct vertices in have exactly common neighbors or none at all. -graphs were introduced and studied by M. Malder in [10]. He proved that in case such graphs are regular. Therefore, -graphs with are Deza graphs. M. Malder proved that if is -regular graph on vertices and has the diameter , then and . In both cases equality is true only for the -dimensional binary cube (the hypercube) when . All -graphs of valency at most 8 was found by A. E. Brouwer in [2] and A. E. Brouwer, P. R. J. Östergård in [3]. In general, the complement of a Deza graph isn’t a Deza graph. Let’s note that the complement of any -graph is a strictly Deza graph.
In [8] W.H. Haemers, H. Kharaghani, and M. Meulenberg introduced and studied a notion of divisible design graphs. A -regular graph on vertices is a divisible design graph (DDG for short) with parameters if the vertex set can be partitioned into classes of size , such that two distinct vertices from the same class have exactly common neighbors, and two vertices from different classes have exactly common neighbors. Divisible design graph with , , or is called improper, otherwise it is called proper. Divisible design graphs are a special case of the notion of Deza graphs. Moreover, Deza graphs with are divisible design graphs (see Proposition 2).
Strongly regular graphs with parameters such that are known (Theorem 1.3.1(v) in [1]). Deza graphs with parameters such that was obtained in the paper [5, Theorem 2.6]. Deza graphs with were characterised in [9] and [6].
In this paper Deza graphs with parameters and are studied.
Let be a graph. If then the set of all neighbors of in we denote by . The set of all vertices at distance precisely from in we denote by . If is a set of vertices of , then is used to denote the union of the neighborhoods of the vertices of .
The paper organized as follows. At first we consider Deza graphs with parameters , where . We consider only connected Deza graphs. If a Deza graph is disconnected, then and Theorem 1 describes its connected components.
Theorem 1
Let be a connected Deza graph with parameters . Then one of the following cases holds:
* is the strictly Deza graph with parameters and it is isomorphic to the -grid.* 2.
* is the Deza graph with parameters and it is isomorphic to the non-incidence graph of the Fano plane;* 3.
* is isomorphic to the four-dimensional binary cube ,* 4.
* has parameters and diameter more than . In particular, if has parameters , then it is isomorphic to the incidence graph of the Fano plane;* 5.
* is the Petersen graph.*
Our next step is to prove that if is a Deza graph with parameters and , then is a DDG. When we have .
Theorem 2
Let be a connected Deza graph with parameters . Then either or is a DDG.
If is a Deza graph with parameters , then one of the following cases holds:
* is one of the strictly Deza graphs with parameters or ;* 2.
* is one of the strongly regular graphs with parameters , or ;* 3.
* has parameters and diameter more than .*
If is a Deza graph with parameters , then one of the following cases holds:
* is isomorphic to the complement of the disjoint union of cubes and it has parameters ;* 2.
* is the Deza graph with parameters and it is isomorphic to the non-incidence graph of the Fano plane;* 3.
* is isomorphic to the four-dimensional cube with parameters .*
In the last section we consider divisible design graphs with parameters , where . In particular, we prove that .
Theorem 3
Let be a DDG with parameters and . Then one of the following statements holds:
* has parameters and it is isomorphic to the non-incidence graph of points and lines of the Fano plane;*
* has parameters and it is isomorphic to the incidence graph of points and lines of the Fano plane;*
* has parameters and it is isomorphic to the -grid.*
2 Case of a=0, a=k-3 or a=k-4
In this section we consider some general properties of Deza graphs. Also we consider Deza graphs with restriction on parameters and , or . Moreover, we get a sufficient condition for a Deza graph to be a divisible design graph.
Let be a strictly Deza graph with parameters . Let also be the number of vertices such that and be the number of vertices such that . It is clear that for any vertex .
Proposition 1
The following equality holds for any vertex :
[TABLE]
* is divided by .*
**Proof. ** It follows from Proposition 1.1 in [5].
Proposition 2
Let be a Deza graph with parameters where then is a DDG.
**Proof. ** Let’s consider a binary relation on . Let be ”coincide or have b common neighbors”. If is equivalence then equivalent classes are classes of the canonical partition of DDG. It is clear that is an equivalence relation if . Suppose that and there are vertices , , such that and . Then and have at least common neighbors in . If then and have common neighbors. So .
Proposition 3
Let be a Deza graph with parameters . Then any connected component of is isomorphic one of the following graphs:
* is the strictly Deza graph with parameters ,* 2.
* is isomorphic to the four-dimensional binary cube ,* 3.
* is the Deza graph with parameters which is isomorphic to the non-incidence graph of the Fano plane,* 4.
* has parameters and diameter more than ,* 5.
* is the Petersen graph.*
**Proof. ** Let be a connected Deza graph with parameters . Consider two cases: graph has a triangle and has no triangles.
Suppose has a triangle and this triangle is induced by , , . Then and have the common neighbor . Since , we have and adjacent with all vertices in exclude one vertex, say . Now we have two cases. If and don’t belong to a triangle, then induces a complete graph. If and belong to a triangle, then induces a complete graph with removed perfect matching.
At first, we consider the case when induces a complete graph. Let’s calculate the number of edges between and . We have edges between and . Also edges between and . Thus, there are edges between and . On the other hand, there are edges between and . Let . Then and it implies that either , or , . It is easy to see that if and then is the strictly Deza graph with parameters and if , then has parameters and diameter more than 2.
The second case, when induces a complete graph with removed perfect matching, is impossible, because vertices and have common neighbors in and is their common neighbor too. It is a contradiction.
Now suppose graph has no triangles. If we calculate edges between and in two ways, then we have equation , where . Since and are mutually prime integers, then divides and this implies that or . If then has parameters . Thus, either is a Petersen graph, or its diameter is greater than 2. If then is the four-dimensional binary cube or is a graph with parameters that is isomorphic to the non-incidence graph of the Fano plane (for more details see [10] and [2, Tables 1, 2]).
Theorem 1 follows from Proposition 3.
Proposition 4
Let be a Deza graph with parameters and . Then one of the following cases holds:
* is one of the strictly Deza graphs with parameters or ,* 2.
* is one of the strongly regular graphs with parameters , or ,*
**Proof. ** By Proposition 1 we have .
Since , then . On the other hand, if we consider non-adjacent vertices we have either or . In any case, , then and . But all such Deza graphs are known [5, 7]. In this case is one of the strictly Deza graphs with parameters or . If is a strongly regular graph, then has parameters or .
Proposition 5
Let be a Deza graph with parameters and . Then is isomorphic to the complement of the disjoint union of cubes for some integer and its parameters are .
**Proof. ** By Proposition 1 we have .
Since , then . On the other hand, if we consider non-adjacent vertices we have either or . In any case, . If then has parameters . The complement of has parameters hence is isomorphic to the disjoint union of some (say ) cubes . So the parameters of are .
If then and . In the case by [5, 7] we don’t have any graphs. In the case we have the parameters sets , , , and . By [10, Theorem 12], there are no Deza graphs with parameters , . In the case and : is even and are odd. It is impossible by Proposition 1.
Now Theorem 2 follows from Propositions 3, 4, and 5.
3 Divisible design graphs
Let be a divisible design graph with parameters . Since is a Deza graph, then it has parameters , where . By Propositions 4, 5, there are no DDGs with parameters and when . Further we consider the case .
Proposition 6
There are no divisible design graphs with parameters , where and .
**Proof. ** Further we prove this proposition in a number of lemmas. Let be a divisible design graphs with parameters , where and .
Lemma 1
Let be a DDG with parameters and . Then the following properties hold:
, 2.
* divides ,* 3.
.
**Proof. ** DDG with parameters has real eigenvalues
[TABLE]
[8, Lemma 2.1]. If then and hence It is a contradiction. Therefore, .
From [8, equation (1)] follows that . Hence, divides .
Since , then is a quadratic residue modulo and .
A partition of the vertices of a graph is equitable if for every pair of indices , which are not necessarily distinct, there is a non-negative integer such that each vertex in has exactly neighbors in , regardless of the choice of .
The vertex partition from the definition of a DDG is called the canonical partition.
Lemma 2
The canonical partition of a proper DDG is equitable.
**Proof. ** This is Theorem 3.1 [8].
Lemma 3
Let be a DDG with parameters then each class of the canonical partition of is a coclique.
**Proof. ** Let be a class of the canonical partition of . Let us consider such that is adjacent to . Since , then and there is the only vertex in . Each common neighbor of and also is a neighbor of . Then
[TABLE]
Hence we have a contradiction to the fact that is a class of the canonical partition of .
If is a set from , then we denote by . Denote by the class of canonical partition of containing a vertex .
Lemma 4
Let be a DDG with parameters and let be a class of canonical partition of . If , then and divides .
**Proof. ** Let . Then adjacent with all vertices from but since the canonical partition of is equitable each vertex from adjacent with all vertices from . Hence, for each vertex we have and divides .
Since then we need to study two cases: and .
Let’s first begin with the case .
Proposition 7
There are no DDGs with parameters , where .
**Proof. ** Let be a DDG with parameters . Then [8, equation 1] implies that
[TABLE]
Hence is an even integer and . Thus, is an odd integer and hence is an even integer. Moreover, the equation
[TABLE]
is hold.
On the other hand, we have
[TABLE]
from [8, equation (2)].
By equation (2) we have . Since is an even integer, then and are odd integers.
Denote by and by . Then and . Let’s put these expressions into equation (1).
Since , then .
Denote by . Then . Since then . Moreover, since and , then
[TABLE]
Since
[TABLE]
then the right part of equation (4) is divided by .
Since and are odd integers, then and has a prime divisor . But can’t be divided by . Then is divided by . Thus, is divided by and can’t be divisor of . It is a contradiction.
Let’s study the case . Let be a DDG with parameters and . Let be a class of the canonical partition of . Let be a set from . Denote by . Since , then . Let’s consider these cases one by one.
Lemma 5
.
**Proof. ** Let . Then the intersection of any two sets and are empty, , .
Let . If has less than neighbors in , then has at least one neighbor in for each . So has neighbors in and at least neighbors out of . It is a contradiction.
Hence, any in has neighbors in and . It follows by lemma 4 that divides . On the other hand, divides by lemma 1. Hence divides . It is a contradiction, because .
Lemma 6
.
**Proof. ** Let . The intersection of any two sets and are empty, , .
Let . If has more than neighbors in then has at least one non-adjacent vertex in each set . It is a contradiction, because has exactly two non-adjacent vertices in . Thus, any vertex in has neighbors in . We have possibilities for choosing in . Hence, . By lemma 4 divides . As in the previous lemma divides . It is a contradiction.
For any two vertices we denote the set by .
Lemma 7
.
**Proof. ** Let . We have exactly three vertices in for any , .
Since , then the remaining two vertices in have no common neighbors in . Thus, one of these two vertices has the only neighbor in .
Let’s consider in , and let be a vertex which has the only neighbor in . Then contains and . Therefore, is contained in .
Since , then and . However, each pairs , , and must have three common neighbors in . It is impossible.
Lemma 8
If , then .
**Proof. ** Let . We have exactly two vertices in for any , and the remaining two vertices in have no common neighbors in . Thus, there are two possibilities for the remaining two vertices in .
- (a)
There is a vertex in without neighbor in . 2. (b)
Each vertex in has the only neighbor in and .
(a) If and has no neighbors in , then vertices , and have exactly common neighbors. Thus, each vertex is adjacent to all vertices in , otherwise . It is impossible by the previous lemmas. Hence, .
(b) Now for any set of four vertices from and for any pair in each vertex in has the only neighbor in and .
Let and for each . For any two different vertices denote the union of all by . Let and . It is significant that is disjoint union of and . Moreover, each vertex from is adjacent to all vertices in , and all vertices exclude one vertex (say ) in , and exactly one vertex (say ) in .
Let . We prove by induction on the following:
For any there is a set of vertices in such that , each vertex from is adjacent to all vertices in , and all vertices exclude the only vertex (say ) in , and one vertex (say ) in .
We have and each vertex has all neighbors from , 3 neighbors in , and . By (b) each vertex from is adjacent to three vertices in . Hence . Thus, . We take this statement as the basis step of induction.
Let’s append vertices one by one to and let be a subset of such that , and . Moreover, for any .
Let’s append to . If , then , , have common neighbors in and have at most common neighbors in . Hence, and we have a contradiction by Lemmas 6, 7. Thus, .
Let . Since and for have common neighbors in , then . If adjacent to , then and have at least common neighbors in . Hence . Let .
Consider the vertices , , and . These four vertices have common neighbors in and common neighbors in . Hence but , and also have common neighbors, because is non-adjacent to vertices in both sets and . Then we have the same situation as in case (a). Hence, . It is a contradiction.
Let . Study the neighbors of in . If is non-adjacent to vertices , then has at least common neighbors ( in , and in , and maybe one in ) with , where . If is non-adjacent to the only vertex , then and has common neighbors in , for any , where . So adjacent to all vertices from . Hence . It is a contradiction.
Thus, adjacent to all vertices from and has one more neighbor outside . Denote this neighbor by . Also denote a vertex from by . Thus, the induction step is proved.
If we add all vertices from to then we have . But divides by lemma 4 and divides by lemma 1. It is a contradiction.
Lemma 9
If , then equals either or .
**Proof. ** By Lemmas 5 – 7 there are no fours vertices in such that . If there is four vertices in for which , then by Lemma 8. Thus, for any four of vertices in we have .
Let . Let be the only vertex in . If is adjacent to a vertex in , then we can find four vertices in such that . Hence, for any fours vertices in and .
Lemma 10
For and one of the following cases holds:
* If then ;*
* If then or .*
**Proof. ** By Lemma 4 divides . By Lemma 1(ii) is divided by . Hence divides in the first case and divides in the second case. This two cases can’t hold simultaneously, because and are mutually prime.
Denote by .
Remark 1
Since the canonical partition of a DDG is equitable (by lemma 2), if , then . This fact implies that each vertex from can’t have more than neighbors in .
Lemma 11
* is a class of the canonical partition of and .*
**Proof. ** By Lemma 10 we have two cases.
Let and . We have and edges from any vertex of to . Therefore, there are exactly edges between and . Moreover, each pair of different vertices of has one common neighbor in . There are pairs of different vertices of . Thus, there are vertices in .
Let and either or . If and , then and any have two common neighbors in . Let’s calculate the number of edges between and in two ways. If , then we have edges between and and at most edges between to . It is a contradiction. Let . We have edges from to and at most edges from to . Then each vertex from has neighbors in . There are edges between and , hence .
Now we can finish our proof of Proposition 6. There is a unique class for each class in . Any two vertices , where from and from , have no common neighbors in . Hence and by Lemma 4 divides . Let and let be a vertex from . Since adjacent to all vertices in , then and has common neighbors in and maybe there are common neighbors of and in . Then and or modulo . It is a contradiction. Thus, there are only classes and in and .
If then we have the same contradiction because or modulo . Therefore, the case is impossible and Proposition 6 is proved.
Let’s prove Theorem 3. Let be a DDG with parameters . If is a disconnected DDG, then by [5, Proposition 4.3] each component of is a strongly regular graph with parameters . There are no such graphs with . If is a connected DDG with , then by Propositions 3 is isomorphic to one of the following graphs: the incidence graph of the Fano plane, the non-incidence graph of the Fano plane , and the complement of the three-dimensional binary cube. By Proposition 6 there are no DDGs with and .
ACKNOWLEDGEMENTS
We are grateful to the Mikhail Lepchinskiy for useful remarks.
ORCID
Vladislav V. Kabanov http://orcid.org/0000-0001-7520-3302
Leonid Salaginov https://orcid.org/0000-0001-6912-2493
REFERENCES
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