The clique graph of a graph \(G\) is the graph whose vertex set is the set of cliques of \(G\) and two vertices are adjacent if and only if the corresponding cliques have non-empty intersection. A graph is self-clique if it is isomorphic to its clique graph. In this paper, we present several results on connected self-clique graphs in which each clique has the same size \(k\) for \(k = 2\) and \(k = 3\).

All parabolic ovals in affine planes of even order \(q \leq 64\) which are preserved by a collineation group isomorphic to \(\mathrm{A\Gamma L}(1,q)\) are determined. They are either parabolas or translation ovals.

We consider the class \({ER}(n, d, \lambda)\) of edge-regular graphs for some \(n > d > \lambda\), i.e., graphs regular of degree \(d\) on \(n\) vertices, with each pair of adjacent vertices having \(\lambda\) common neighbors. It has previously been shown that for such graphs with \(\lambda > 0\) we have \(n \geq 3(d – \lambda)\) and much has been done to characterize such graphs when equality holds.

Here we show that \(n \geq 3(d – \lambda) + 1\) if \(\lambda > 0\) and \(d\) is odd and contribute to the characterization of the graphs in \({ER}(n, d, \lambda)\), \(\lambda > 0\), \(n = 3(d-\lambda)+1\) by proving some lemmas about the structure of such graphs, and by classifying such graphs that satisfy a strong additional requirement, that the number \(t = t(u,v)\) of edges in the subgraph induced by the \(\lambda\) common neighbors of any two adjacent vertices \(u\) and \(v\) is positive, and independent of \(u\) and \(v\). The result is that there are exactly 4 such graphs: \(K_4\) and 3 strongly regular graphs.

If \(G\) is a connected graph, the distance \(d(u, v)\) between two vertices \(u,v \in V(G)\) is the length of a shortest path between them. Let \(W = \{w_1, w_2, \ldots, w_k\}\) be an ordered set of vertices of \(G\) and let \(v\) be a vertex of \(G\). The representation \(r(v|W)\) of \(v\) with respect to \(W\) is the \(k\)-tuple \((d(v, w_1), d(v, w_2), \ldots, d(v, w_k))\). If distinct vertices of \(G\) have distinct representations with respect to \(W\), then \(W\) is called a resolving set or locating set for \(G\). A resolving set of minimum cardinality is called a basis for \(G\) and this cardinality is the metric dimension of \(G\), denoted by \(\dim(G)\).

A family \(\mathcal{G}\) of connected graphs is a family with constant metric dimension if \(\dim(G)\) does not depend upon the choice of \(G\) in \(\mathcal{G}\). In this paper, we are dealing with the study of metric dimension of Möbius ladders. We prove that Möbius ladder \(M_n\) constitute a family of cubic graphs with constant metric dimension and only three vertices suffice to resolve all the vertices of Möbius ladder \(M_n\), except when \(n \equiv 2 \pmod{8}\). It is natural to ask for the characterization of regular graphs with constant metric dimension.

In this paper, we obtain an interesting identity by applying two \(g\)-operator identities. From this identity, we can recover the terminating Sears’ \(\prescript{}{3}{\Phi}_2\) transformation formulas and the Dilcher’s identity and the Uchimura’s identity. In addition, an interesting binomial identity can be concluded.

Let \(G\) be a connected graph. For \(x,y \in V(G)\) with \(d(x,y) = 2\), we define \(J(x,y) = \{u \in N(x) \cap N(y) | N[u] \cap N[x] \cup N[y]\}\) and \(J'(x,y) = \{u \in N(x) \cap N(y) |\) if \(v \in N(u) \setminus (N[x] \cup N[y])\) then \(N(x) \cup N(y) \cup N(u) \cap N[v]\}\). A graph \(G\) is quasi-claw-free if \(J(x,y) \neq \emptyset\) for each pair \((x,y)\) of vertices at distance \(2\) in \(G\). Broersma and Vumar introduced the class of \(P_3\)-dominated graphs defined as \(J(x,y) \cup J'(x,y) \neq \emptyset\) for each \(x,y \in V(G)\) with \(d(x,y) = 2\). Let \(\kappa(G)\) and \(\alpha_2(G)\) be the connectivity of \(G\) and the maximum number of vertices that are pairwise at distance at least \(2\) in \(G\), respectively. A cycle \(C\) is \(m\)-dominating if \(d(x,C) = \min\{d(x,u) | u \in V(C)\} \leq m\) for all \(x \in V(G)\). In this note, we prove that every \(2\)-connected \(\mathcal{P}_3\)-dominated graph \(G\) has an \(m\)-dominating cycle if \(\alpha_{2m+3}(G) \leq \kappa(G)\).

We initiate the study of signed edge majority total domination in graphs. The open neighborhood \(N_G(e)\) of an edge \(e\) in a graph \(G\) is the set consisting of all edges having a common vertex with \(e\). Let \(f\) be a function on \(E(G)\), the edge set of \(G\), into the set \(\{-1, 1\}\). If \(\sum_{x \in N_G(e)} f(x) \geq 1\) for at least half of the edges \(e \in E(G)\), then \(f\) is called a signed edge majority total dominating function of \(G\). The value \(\sum_{e\in E(G)}f(e)\), taking the minimum over all signed edge majority total dominating functions \(f\) of \(G\), is called the signed edge majority total domination number of \(G\) and denoted by \(\gamma’_{smt}(G)\). Obviously, \(\gamma’_{smt}(G)\) is defined only for graphs \(G\) which have no connected components isomorphic to \(K_2\). In this paper, we establish lower bounds on the signed edge majority total domination number of forests.

This article is a contribution to the study of the automorphism groups of \(2\)-\((v,k,1)\) designs. Let \(\mathcal{D}\) be a \(2\)-\((v,13,1)\) design, \(G \leq \mathrm{Aut}(\mathcal{D})\) be block transitive and point primitive. If \(G\) is unsolvable, then \(\mathrm{Soc}(G)\), the socle of \(G\), is not \(\mathrm{Sz}(q)\).

Using Cioaba’s inequality on the sum of the 3rd powers of the vertex degrees in connected graphs, we present an inequality on the Laplacian eigenvalues of connected graphs.

In this paper, the author studies the relation of vertices, edges, and cells of the quasi-cross-cut partition. Moreover, the three-term recurrence relations of \(\dim(S_d^0(\Delta))\) over the quasi-cross-cut partition and the triangulation are presented.