Related papers: There exists no distance-regular graph with inters…

A 4-regular matchstick graph is a planar unit-distance graph whose vertices have all degree 4. Examples of 4-regular matchstick graphs are currently known for all number of vertices $\geq$ 52 except for 53, 55, 56, 58, 59, 61, and 62. In…

In this note we obtain an explicit formula for the Hosoya polynomial of any distance-regular graph in terms of its intersection array. As a consequence, we obtain a very simple formula for the Hosoya polynomial of any strongly regular…

All finite simple self $2$-distance graphs with no $4$-cycle, diamond, or triangles with a common vertex are determined. Utilizing these results, it is shown that there is no cubic self $2$-distance graphs.

We show that there is no $(95,40,12,20)$ strongly regular graph and, consequently, there is no $(96,45,24,18)$ strongly regular graph, no two-graph on $96$ vertices, and no partial geometry $\rm{pg}(5,9,3)$. The main idea of the result is…

Highly-regular graphs can be regarded as a combinatorial generalization of distance-regular graphs. From this standpoint, we study combinatorial aspects of highly-regular graphs. As a result, we give the following three main results in this…

We introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. Let $\Gamma$ be a graph with vertex set $V$, diameter $D$, adjacency matrix $A$, and…

In this article, we study quantum automorphism groups of distance-transitive graphs. We show that the odd graphs, the Hamming graphs $H(n,3)$, the Johnson graphs $J(n,2)$ and the Kneser graphs $K(n,2)$ do not have quantum symmetry. We also…

We consider the problem of which distance-regular graphs with small valency are Cayley graphs. We determine the distance-regular Cayley graphs with valency at most $4$, the Cayley graphs among the distance-regular graphs with known putative…

Let $G=(V,E)$ be a finite, simple, connected, combinatorial graph on $n$ vertices and let $D \in \mathbb{R}^{n \times n}$ be its graph distance matrix $D_{ij} = d(v_i, v_j)$. Steinerberger (J. Graph Theory, 2023) empirically observed that…

We present a proof, using spectral techniques, that there is no finite measurable coloring of the odd-distance graph.

Maximum size of equiangular lines in $\mathbb{R}^{19}$ has been known in the range between 72 to 76 since 1973. Acoording to the nonexistence of strongly regular graph $(75,32,10,16)$ \cite{aza15}, Larmen-Rogers-Seidel Theorem \cite{lar77}…

We consider a distance-regular graph $\G$ with diameter $d \ge 3$ and eigenvalues $k=\theta_0>\theta_1>... >\theta_d$. We show the intersection numbers $a_1, b_1$ satisfy $$ (\theta_1 + {k \over a_1+1}) (\theta_d + {k \over a_1+1}) \ge -…

A non-complete \drg $\Gamma$ is called geometric if there exists a set $\mathcal{C}$ of Delsarte cliques such that each edge of $\Gamma$ lies in a unique clique in $\mathcal{C}$. In this paper, we determine the non-complete distance-regular…

It is easy to see that in a connected graph any 2 longest paths have a vertex in common. For k>=7, Skupien in [7] obtained a connected graph in which some k longest paths have no common vertex, but every k-1 longest paths have a common…

The concept of pseudo-distance-regularity around a vertex of a graph is a natural generalization, for non-regular graphs, of the standard distance-regularity around a vertex. In this note, we prove that a pseudo-distance-regular graph…

Edge-girth-regular graphs (abbreviated as \emph{egr} graphs) are regular graphs in which every edge is contained in the same number of shortest cycles. We prove that there is no $3$-regular \emph{egr} graph with girth $7$ such that every…

We prove that there is no strongly regular graph (SRG) with parameters (460,153,32,60). The proof is based on a recent lower bound on the number of 4-cliques in a SRG and some applications of Euclidean representation of SRGs.

A graph $G$ is a link-irregular graph if every two distinct vertices of $G$ have non-isomorphic links. The link of a vertex $v$ in $G$ is the subgraph induced by the neighbors of $v$ in $G$. Ali, Chartrand and Zhang [Discussiones…

The first part (page 1 - 7) of this article presents the currently known examples of 4-regular matchstick graphs with 63 - 70 vertices. The second part (page 8 - 15) presents the currently known examples of $(2;4)$-regular matchstick graphs…

We construct distance-regular graphs, including strongly regular graphs, admitting a transitive action of the Chevalley groups $G_2(4)$ and $G_2(5)$, the orthogonal group $O(7,3)$ and the Tits group $T=$$^2F_4(2)'$. Most of the constructed…