This paper outlines a method for constructing self-orthogonal codes from orbit matrices of strongly regular graphs admitting an automorphism group $G$ which acts with orbits of length $w$, where $w$ divides $|G|$. We apply this method to construct self-orthogonal codes from orbit matrices of the strongly regular graphs with at most 40 vertices. In particular, we construct codes from adjacency or orbit matrices of graphs with parameters $(36, 15, 6, 6)$, $(36, 14, 4, 6)$, $(35, 16, 6, 8)$ and their complements, and from the graphs with parameters $(40, 12, 2, 4)$ and their complements. That completes the classification of self-orthogonal codes spanned by the adjacency matrices or orbit matrices of the strongly regular graphs with at most 40 vertices. Furthermore, we construct ternary codes of $2$-$(27,9,4)$ designs obtained as residual designs of the symmetric $(40, 13, 4)$ designs (complementary designs of the symmetric $(40, 27, 18)$ designs), and their ternary hulls. Some of the obtained codes are optimal, and some are best known for the given length and dimension.

One of the main problems in the theory of strongly regular graphs (SRGs) is constructing and classifying SRGs with given parameters. Strongly regular graphs with parameters $(37,18,8,9)$, $(41,20,9,10)$, $(45,22,10,11)$, $(49,24,11,12)$, $(49,18,7,6)$ and $(50,21,8,9)$ are the only strongly regular graphs on up to 50 vertices that still have to be classified. In this paper, we give the enumeration of SRGs with these parameters having $S_3$ as an automorphism group. The construction of SRGs in this paper is a step in the classification of SRGs on up to 50 vertices.

Consider a bipartite distance-regularized graph \(\Gamma \) with color partitions Y and \(Y'\). Notably, all vertices in partition Y (and similarly in \(Y'\)) exhibit a shared eccentricity denoted as D (and \(D'\), respectively). The characterization of bipartite distance-regularized graphs, specifically those with \(D \le 3\), in relation to the incidence structures they represent is well established. However, when \(D=4\), there are only two possible scenarios: either \(D'=3\) or \(D'=4\). The instance where \(D=4\) and \(D'=3\) has been previously investigated. In this paper, we establish a one-to-one correspondence between the incidence graphs of quasi-symmetric SPBIBDs with parameters \((v, b, r, k, \lambda _1, 0)\) of type \((k-1, t)\), featuring intersection numbers \(x=0\) and \(y>0\) (where \(y \le t < k\)), and bipartite distance-regularized graphs with \(D=D'=4\). Moreover, our investigations result in the systematic classification of 2-Y-homogeneous bipartite distance-regularized graphs, which are incidence graphs of quasi-symmetric SPBIBDs with parameters \((v,b,r,k, \lambda _1,0)\) of type \((k-1,t)\) with intersection numbers \(x=0\) and \(y=1\).

Regular two-graphs on up to 36 vertices are classified, and recently, the classification of regular two-graphs on 38 and 42 vertices having at least one descendant with a nontrivial automorphism group has been performed. The first unclassified cases are those on 46 and 50 vertices. It is known that there are at least 97 regular two-graphs on 46 vertices leading to 2104 descendants and 54 regular two-graphs on 50 vertices leading to 785 descendants. In this paper, we classified all strongly regular graphs with parameters $(45,22,10,11)$, $(49,24,11,12)$, and $(50,21,8,9)$ that have $Z_6$ as the automorphism group and constructed regular two-graphs from SRGs $(45,22,10,11)$, SRGs $(49,24,11,12)$, and SRGs $(50,21,8,9)$ that have automorphisms of order six. In this way, we enumerated all regular two-graphs on up to 50 vertices that have at least one descendant with an automorphism group of order six or at least one strongly regular graph associated with an automorphism group of order six. We found 236 new regular two-graphs on 46 vertices leading to 3172 new SRG $(45,22,10,11)$ and 51 new regular two-graphs on 50 vertices leading to 398 new SRG $(49,24,11,12)$.

Distance-biregular graphs form a natural bipartite generalization of distance-regular graphs. Among them, the class of 2- Y -homogeneous distance-biregular graphs plays a prominent role and has been the subject of several recent classification efforts. In this paper, we complete the classification of 2- Y -homogeneous $$(Y,Y')$$ ( Y , Y ′ ) -distance-biregular graphs with eccentricity $$D=4$$ D = 4 . Building on earlier work that settled the cases $$c_2'=1$$ c 2 ′ = 1 and $$c_2'=2$$ c 2 ′ = 2 , we address the remaining open case $$c_2'\ge 3$$ c 2 ′ ≥ 3 . We prove that no such graphs exist, thereby resolving an open problem posed in previous work and closing the classification program for eccentricity $$D=4$$ D = 4 . Our approach combines a detailed analysis of intersection numbers, arithmetic constraints arising from 2- Y -homogeneity, and structural properties of distance-biregular graphs. As a consequence, we obtain a complete characterization of all 2- Y -homogeneous distance-biregular graphs with $$D=4$$ D = 4 , and we further conclude that every 2- Y -homogeneous distance-biregular graph with $$c_2'\ge 3$$ c 2 ′ ≥ 3 must necessarily have eccentricity $$D=3$$ D = 3 .

In 2003, Harada and Tonchev presented a construction of self-orthogonal codes from orbit matrices of symmetric $2$-designs with fixed point free automorphisms. Since then, the constructions of self-orthogonal codes from orbit matrices of $2$-designs has been extensively studied. In this paper, we present new constructions of self-orthogonal codes from orbit matrices of $2$-designs for the cases not covered by the previously described methods. We construct self-orthogonal codes from orbit matrices of $2$-$(1024,496,249)$ and $2$-$(45,5,1)$ designs. Some of the constructed codes are optimal.