In this paper, we introduce the notion of \((\alpha, \beta)\)-generalized \(d\)-derivations on lattices and investigate some related properties. Also, using the notion of permuting \((\alpha, \beta)\)-triderivation, we characterize the distributive elements of a lattice.

Suppose \(\{P_r\}\) is a nonempty family of paths for \(r \geq 3\), where \(P_r\) is a path on \(r\) vertices. An \(r\)-coloring of a graph \(G\) is said to be \(\{P_r\}\)-free if \(G\) contains no 2-colored subgraph isomorphic to any path \(P_r\) in \(\{P_r\}\). The minimum \(k\) such that \(G\) has a \(\{P_r\}\)-free coloring using \(k\) colors is called the \(\{P_r\}\)-free chromatic number of \(G\) and is denoted by \(\chi_{\{P_r\}}(G)\). If the family \(\{P_r\}\) consists of a single graph \(P_r\), then we use \(\chi_{P_r}(G)\). In this paper, \(\{P_r\}\)-free colorings of Sierpiński-like graphs are considered. In particular, \(\chi_{P_3}(S_n)\), \(\chi_{P_4}(S_n)\), \(\chi_{P_4}(S(n, k))\), \(\chi_{P_3}(S^{++}(n, k))\), and \(\chi_{P_4}(S^{++}(n, k))\) are determined.

Let \(G = (V,E)\) be a graph with \(v = |V(G)|\) vertices and \(e = |E(G)|\) edges. An \((a, d)\)-edge-antimagic total labeling of the graph \(G\) is a one-to-one map \(A\) from \(V(G) \cup E(G)\) onto the integers \(\{1,2,\ldots,v+e\}\) such that the set of edge weights of the graph \(G\), \(W = \{w(xy) : xy \in E(G)\}\) form an arithmetic progression with the initial term \(a\) and common difference \(d\), where \(w(xy) =\lambda(x) + \lambda(y) + \lambda(xy)\) for any \(xy \in E(G)\). If \(\lambda(V(G)) = \{1,2,\ldots,v\}\) then \(G\) is super \((a, d)\)-edge-antimagic total, i.e., \((a,d)\)-EAT. In this paper, for different values of \(d\), we formulate super \((a, d)\)-edge-antimagic total labeling on subdivision of stars \(K_{1,p}\) for \(p \geq 5\).

We discuss the chromaticity of one family of \(K_4\)-homeomorphs which has girth \(7\) and has exactly \(1\) path of length \(1\), and give a sufficient and necessary condition for the graphs in the family to be chromatically unique.

A theta graph is denoted by \(\theta(a,b,c)\), where \(a \leq b \leq c\). It is obtained by subdividing the edges of the multigraph consisting of \(3\) parallel edges \(a\) times, \(b\) times, and \(c\) times each. In this paper, we show that the theta graph is matching unique when \(a \geq 2\) or \(a = 0\), and all theta graphs are matching equivalent when only one of the edges is subdivided one time. We also completely characterize the relation between the largest matching root \(\alpha\) and the length of path \(a, b, c\) of a theta graph, and determine the extremal theta graphs.

The line graph of \(G\), denoted \(L(G)\), is the graph with vertex set \(E(G)\), where vertices \(x\) and \(y\) are adjacent in \(L(G)\) if and only if edges \(x\) and \(y\) share a common vertex in \(G\). In this paper, we determine all graphs \(G\) for which \(L(G)\) is a circulant graph. We will prove that if \(L(G)\) is a circulant, then \(G\) must be one of three graphs: the complete graph \(K_4\), the cycle \(C_n\), or the complete bipartite graph \(K_{a,b}\), for some \(a\) and \(b\) with \(\gcd(a,b) = 1\).

Let \(G\) be a graph. The point arboricity of \(G\), denoted by \(\rho (G)\), is the minimum number of colors that can be used to color the vertices of \(G\) so that each color class induces an acyclic subgraph of \(G\). The list point arboricity \(\rho_l(G)\) is the minimum \(k\) so that there is an acyclic \(L\)-coloring for any list assignment \(L\) of \(G\) which \(|L(v)| \geq k\). So \(\rho(G) \leq \rho_l(G)\). Zhen and Wu conjectured that if \(|V(G)| \leq 3\rho (G)\), then \(\rho_l(G) = p(G)\). Motivated by this, we investigate the list point arboricity of some complete multi-partite graphs of order slightly larger than \(3p(G)\), and obtain \(\rho(K_{m,(1),2(n-1)}) = \rho_l(K_{m(1),2(n-1)})\) \((m = 2,3,4)\).

In this paper, we consider the relationship between toughness and the existence of \([a, b]\)-factors. We obtain that a graph \(G\) has an \([a, b]\)-factor if \(t(G) \geq {a-1} + \frac{a-1}{b}\) with \(b > a > 1\). Furthermore, it is shown that the result is best possible in some sense.

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.

It has been known for at least \(2500\) years that mathematics and music are directly related. This article explains and extends ideas originating with Euler involving labeling parts of graphs with notes in such a way that other parts of the graphs correspond in a natural way to chords. The principal focus of this research is the notion of diatonic labelings of cubic graphs, that is, labeling the edges with pitch classes in such a way that vertices are incident with edges labeled with the pitch classes of a triad in a given diatonic scale. The pitch classes are represented in a natural way with elements of \(\mathbb{Z}_{12}\), the integers modulo twelve.

Several classes of cubic graphs are investigated and shown to be diatonic. Among the graphs considered are Platonic Solids, cylinders, and Generalized Petersen Graphs. It is shown that there are diatonic cubic graphs on \(n\) vertices for even \(n \geq 14\). Also, it is shown that there are cubic graphs on \(n\) vertices that do not have diatonic labelings for all even \(n \geq 4\). The question of forbidden subgraphs is investigated, and a forbidden subgraph for diatonic graphs, or “clash”, is demonstrated.

In this paper, we recall Konhauser polynomials. Approximation properties of these operators are obtained with the help of the Korovkin theorem. The order of convergence of these operators is computed by means of modulus of continuity, Peetre’s K-functional, and the elements of the Lipschitz class. Also, we introduce the \(r\)-th order generalization of these operators and we evaluate this generalization by the operators defined in this paper. Finally, we give an application to differential equations.

A graph \(X\) is said to be End-regular (resp., End-orthodox, End-inverse) if its endomorphism monoid \(\mathrm{End}(X)\) is a regular (resp., orthodox, inverse) semigroup. In this paper, End-regular (resp., End-orthodox, End-inverse) graphs which are the join of split graphs \(X\) and \(Y\) are characterized. It is also proved that \(X + Y\) is never End-inverse for any split graphs \(X\) and \(Y\).

Some new families of complete caps in Galois affine spaces \({AG}(N,q)\) of dimension \(N \equiv 0 \pmod{4}\) and odd order \(q \leq 127\) are constructed. No smaller complete caps appear to be known.

We give two Frankl-like results on set systems with restrictions on set difference sizes and set symmetric difference sizes modulo prime powers. Based on a similar method, we also give a bound on codes satisfying the properties of Hamming distance modulo prime powers.

In this note, a resolvable \((K_4 – e)\)-design of order \(296\) is constructed. Combining the results of \([2, 3, 4]\), the existence spectrum of resolvable \((K_4 – e)\)-designs of order \(v\) is the set \(\{v : v \equiv 16 \pmod{20}, v \geq 16\}\).

We study permutations of the set \([n] = \{1, 2, \ldots, n\}\) written in cycle notation, for which each cycle forms an increasing or decreasing interval of positive integers. More generally, permutations whose cycle elements form arithmetic progressions are considered. We also investigate the class of generalized interval permutations, where each cycle can be rearranged in increasing order to form an interval of consecutive positive integers.

In this paper, we study the symmetry for the generalized twisted Genocchi polynomials and numbers. We give some interesting identities of the power sums and the generalized twisted Genocchi polynomials using the symmetric properties for the \(p\)-adic invariant \(q\)-integral on \(\mathbb{Z}_p\).

In this paper, we use a simple method to derive different recurrence relations on the recursive sequence order-\(k\) and their sums, which are more general than that given in literature [J.Feng, More Identities on the Tribonacci Numbers, Ars Combinatoria, \(100(2011), 73-78]\). By using the generating matrices, we get more identities on the recursive sequence order-\(k\) and their sums, which are more general than that given in literature [E.Kihg, Tribonacci Sequences with Certain Indices and Their Sums, Ars Combinatoria, \(86(2008), 13-22]\) .

By applying discharging methods and properties of critical graphs, we proved that every simple planar graph \(G\) with \(\Delta(G) \geq 5\) is of class 1, if any 4-cycle is not adjacent to a 5-cycle in \(G\).

A graph \(G\) is pancyclic if it contains a cycle of every length from 3 to \(|V(G)|\) inclusive. A graph \(G\) is panconnected if there exists a path of length \(l\) joining any two different vertices \(x\) and \(y\) with \(d_G(x,y) \leq l \leq |V(G)| – 1\), where \(d_G(x,y)\) denotes the distance between \(x\) and \(y\) in \(G\). A hamiltonian graph \(G\) is panpositionable if for any two different vertices \(x\) and \(y\) of \(G\) and any integer \(k\) with \(d_G(x,y) \leq k \leq |V(G)|/2\), there exists a hamiltonian cycle \(C\) of \(G\) with \(d_C(x,y) = k\), where \(d_C(x,y)\) denotes the distance between \(x\) and \(y\) in a hamiltonian cycle \(C\) of \(G\). It is obvious that panconnected graphs are pancyclic, and panpositionable graphs are pancyclic.

The above properties can be studied in bipartite graphs after some modification. A graph \(H = (V_0 \cup V_1, E)\) is bipartite if \(V(H) = V_0 \cup V_1\) and \(E(H)\) is a subset of \(\{(u,v) | u \in V_0 \text{ and } v \in V_1\}\). A graph is bipancyclic if it contains a cycle of every even length from 4 to \(2\lfloor |V(H)|/2 \rfloor\) inclusive. A graph \(H\) is bipanconnected if there exists a path of length \(l\) joining any two different vertices \(x\) and \(y\) with \(d_H(x,y) \leq l \leq |V(H)| – 1\), where \(d_H(x,y)\) denotes the distance between \(x\) and \(y\) in \(H\) and \(l – d_H(x,y)\) is even. A hamiltonian graph \(H\) is bipanpositionable if for any two different vertices \(x\) and \(y\) of \(H\) and for any integer \(k\) with \(d_H(x,y) \leq k \leq |V(H)|/2\), there exists a hamiltonian cycle \(C\) of \(H\) with \(d_C(x,y) = k\), where \(d_C(x,y)\) denotes the distance between \(x\) and \(y\) in a hamiltonian cycle \(C\) of \(H\) and \(k – d_H(x,y)\) is even. It can be shown that bipanconnected graphs are bipancyclic, and bipanpositionable graphs are bipancyclic.

In this paper, we present some examples of pancyclic graphs that are neither panconnected nor panpositionable, some examples of panconnected graphs that are not panpositionable, and some examples of graphs that are panconnected and panpositionable, for nonbipartite graphs. Corresponding examples for bipartite graphs are discussed. The existence of panpositionable (or bipanpositionable, resp.) graphs that are not panconnected (or bipanconnected, resp.) is still an open problem.

In \([2]\) Stefano Innamorati and Mauro Zannetti gave a characterization of the planes secant to a non-singular quadric in \({P}G(4, q)\). Their result is based on a particular hypothesis (which we call “polynomial”) that, as the same authors wrote at the end of the paper, could not exclude possible sporadic cases. In this paper, we improve their result by giving a characterization without the “polynomial” hypothesis. So, possible sporadic cases are definitely excluded.

This paper generalizes the results of Guiduli [B. Guiduli, On incidence coloring and star arboricity of graphs. Discrete Math. \(163
(1997), 275-278]\) on the incidence coloring of graphs to the fractional incidence coloring. Tight asymptotic bounds analogous to Guiduli’s results are given for the fractional incidence chromatic number of graphs. The fractional incidence chromatic number of circulant graphs is studied. Relationships between the \(k\)-tuple incidence chromatic number and the incidence chromatic number of the direct products and lexicographic products of graphs are established. Finally, for planar graphs \(G\), it is shown that if \(\Delta(G) \neq 6\), then \(\chi_i(G) \leq \Delta(G) + 5\); if \(\Delta(G) = 6\), then \(\chi_i(G) \leq \Delta(G) + 6\); where \(\chi_i(G)\) denotes the incidence chromatic number of \(G\). This improves the bound \(\chi_i(G) \leq \Delta(G) + 7\) for planar graphs given in [M. Hosseini Dolama, E. Sopena, X. Zhu, Incidence coloring of k-degenerated graphs, Discrete Math. \(283 (2004)\), no. \(1-3, 121-128]\).

Let \(P(G, \lambda)\) be the chromatic polynomial of a graph \(G\). A graph \(G\) is chromatically unique if for any graph \(H\), \(P(H, \lambda) = P(G, \lambda)\) implies \(H \cong G\). Some sufficient conditions guaranteeing that certain complete tripartite graph \(K(l, n, r)\) is chromatically unique were obtained by many scholars. Especially, in 2003, H.W. Zou showed that if \(n > \frac{1}{3}(m^2+k^2+mk+2\sqrt{m^2 + k^2 + mk} + m – k)\), where \(n, k\), and \(m\) are non-negative integers, then \(K(n – m, n, n + k)\) is chromatically unique (or simply \(\lambda\)-unique). In this paper, we show that for any positive integers \(n, m\), and \(k\), let \(G = K(n – m, n, n + k)\), where \(m \geq 2\) and \(k \geq 1\), if \(n \geq \max\{\lceil \frac{1}{4}m^2 + m + k \rceil, \lceil \frac{1}{4}m^2 + \frac{3}{2}m + 2k – \frac{11}{4} \rceil, \lceil mk + m – k + 1 \rceil\}\), then \(G\) is \(\chi\)-unique. This improves upon H.W. Zou’s result in the case \(m \geq 2\) and \(k \geq 1\).

In this paper, it is proved that a toroidal graph without cycles of length \(k\) for each \(k \in \{4, 5, 7, 10\}\) is \(3\)-choosable.

In this paper, we investigate the transitive Cayley graphs of strong semilattices of rectangular groups, and of normal bands, respectively. We show under which conditions they enjoy the property of automorphism vertex transitivity in analogy to Cayley graphs of groups.

A family of connected graphs \(\mathcal{G}\) is said to be a family with constant metric dimension if its metric dimension is finite and does not depend upon the choice of \(G\) in \(\mathcal{G}\). In this paper, we study the metric dimension of the generalized Petersen graphs \(P(n,m)\) for \(n = 2m+1\) and \(m \geq 1\) and give a partial answer to the question raised in \([9]\): Is \(P(n, m)\) for \(n \geq 7\) and \(3 \leq m \leq \lfloor \frac{n-1}{2} \rfloor\) a family of graphs with constant metric dimension? We prove that the generalized Petersen graphs \(P(n,m)\) with \(n = 2m +1\) have metric dimension \(3\) for every \(m \geq 2\).

Let \(G\) be a graph on \(n\) vertices. \(\delta\) and \(\alpha\) be the minimum degree and independence number of \(G\), respectively. We prove that if \(G\) is a \(2\)-connected graph and \(|N(x) \cup N(y)| \geq n-\delta – 1\) for each pair of nonadjacent vertices \(x,y\) with \(1 \leq |N(x) \cap N(y)| \leq \alpha – 1\), then \(G\) is hamiltonian or \(G \in \{G_1, G_2\}\) (see Figure 1.1 and Figure 1.2). As a corollary, if \(G\) is a 2-connected graph and \(|N(x) \cup N(y)| \geq n – \delta\) for each pair of nonadjacent vertices \(x,y\) with \(1 \leq |N(x) \cap N(y)| \leq \alpha – 1\), then \(G\) is hamiltonian. This result extends former results by Faudree et al. \(([5])\) and Yin \(([7])\).

Arising from the VLSI design and network communication, the cutwidth problem for a graph \(G\) is to embed \(G\) into a path such that the maximum number of overlap edges (i.e., the congestion) is minimized. The characterization of forbidden subgraphs or critical graphs is meaningful in the study of a graph-theoretic parameter. This paper characterizes the set of \(4\)-cutwidth critical trees by twelve specified ones.

A path \(P\) in an edge-colored graph (not necessarily a proper edge-coloring) is a rainbow path if no two edges of \(P\) are assigned the same color. For a connected graph \(G\) with connectivity \(\kappa(G)\) and an integer \(k\) with \(1 \leq k \leq \kappa(G)\), the rainbow \(k\)-connectivity \(rc_k(G)\) of \(G\) is the minimum number of colors needed in an edge-coloring of \(G\) such that every two distinct vertices \(u\) and \(v\) of \(G\) are connected by at least \(k\) internally disjoint \(u-v\)rainbow paths. In this paper, the rainbow \(2\)-connectivity of the Petersen graph as well as the rainbow connectivities of all cubic graphs of order \(8\) or less are determined.

This paper investigates the number of rooted simple bipartite maps on the sphere and presents some formulae for such maps with the number of edges and the valency of the root-face as two parameters.

For a graph \(G = (V(G), E(G))\), the transformation graph \(G^{+-+}\) is the graph with vertex set \(V(G) \cup E(G)\) in which the vertices \(\alpha\) and \(\beta\) are joined by an edge if and only if \(\alpha\) and \(\beta\) are adjacent or incident in \(G\) while \(\{\alpha, \beta\} \not\subseteq E(G)\), or \(\alpha\) and \(\beta\) are not adjacent in \(G\) while \(\{\alpha, \beta\} \in E(G)\). In this note, we show that all but for a few exceptions, \(G^{+-+}\) is super-connected and super edge-connected.

In this paper, we give matrix representations of the \(k\)-generalized order-\(k\) Perrin numbers and we obtain relationships between these sequences and matrices. In addition, we calculate the determinant of this matrix.

A graph \(G\) is \(k\)-total domination edge critical, abbreviated to \(k\)-critical if confusion is unlikely, if the total domination number \(\gamma_t(G)\) satisfies \(\gamma_t(G) = k\) and \(\gamma_t(G + e) < \gamma_t(G)\) for any edge \(e \in E(\overline{G})\).Graphs that are \(4\)-critical have diameter either \(2\), \(3\), or \(4\). In previous papers, we characterized structurally the \(4\)-critical graphs with diameter four and found bounds on the order of \(4\)-critical graphs with diameter two. In this paper, we study a family \(\mathcal{H}\) of \(4\)-critical graphs with diameter three, in which every vertex is a diametrical vertex, and every diametrical pair dominates the graph. We also generalize the self-complementary graphs and show that these graphs provide a special case of the family \(\mathcal{H}\).

A finite planar set is \(k\)-isosceles for \(k \geq 3\) if every \(k\)-point subset of the set contains a point equidistant from two others. There exists no convex \(4\)-isosceles \(8\)-point set with \(8\) points on a circle.

In this note, motivated by the non-existence of a vertex-transitive strongly regular graph with parameters \((3250, 57, 0, 1)\), we obtain a feasibility condition concerning strongly regular graphs admitting an automorphism group with exactly two orbits on vertices. We also establish a result on the possible orbit sizes of a potential strongly regular graph with parameters \((3250, 57, 0, 1)\). We use our results to obtain a list of only 11 possible orbit size combinations for a potential strongly regular graph with parameters \((3250, 57, 0, 1)\) admitting an automorphism group with exactly two orbits.

Let \(G = (V, E)\) be a simple connected graph, where \(d_u\) is the degree of vertex \(u\), and \(d_G(u, v)\) is the distance between \(u\) and \(v\). The Schultz index of \(G\) is defined as \(\mathcal{W}_+(G) = \sum\limits_{u,v \subset V(G)} (d_u + d_v)d_G(u,v).\)In this paper, we investigate the Schultz index of a class of trees with diameter not more than \(4\).

Let \(G\) be a graph with a maximum matching of size \(q\), and let \(p \leq q\) be a positive integer. Then \(G\) is called \((p, q)\)-extendable if every set of \(p\) independent edges can be extended to a matching of size \(q\). If \(G\) is a graph of even order \(n\) and \(n = 2q\), then \((p,q)\)-extendable graphs are exactly the \(p\)-extendable graphs defined by Plummer \([11]\) in \(1980\).

Let \(d \geq 3\) be an integer, and let \(G\) be a \(d\)-regular graph of order \(n\) with a maximum matching of size \(q = \frac{n-t}{2}\geq 3\) for an integer \(t \geq 1\) such that \(n – t\) is even. In this work, we prove that if

(i) \(n \leq {(t+4)(d+1)-5}\) or

(ii) \(n \leq (t+4)(d+2) – 1\) when \(d\) is odd,

then \(G\) is \((2, q)\)-extendable.

A graph with vertex set \(V\) is said to have a prime cordial labeling if there is a bijection \(f\) from \(V\) to \(\{1,2,\ldots,|V|\}\) such that if each edge \(uv\) is assigned the label \(1\) for the greatest common divisor \(\gcd(f(u), f(v)) = 1\) and \(0\) for \(\gcd(f(u), f(v)) = 1\), then the number of edges labeled with \(0\) and the number of edges labeled with \(1\) differ by at most \(1\). In this paper, we show that the Flower Snark and its related graphs are prime cordial for all \(n \geq 3\).

In [2] it is proved that if \(X = Cay(G, S)\) is a connected tetravalent Cayley graph on a regular \(p\)-group \(G\) (for \(p \neq 2, 5\)), then the right regular representation of \(G\) is normal in the automorphism group of \(X\). In this paper, we prove that a similar result holds, for \(p = 5\), under a slightly stronger hypothesis. Some remarkable examples are presented.

In this paper, we define, for a graph invariant \(\psi\), the deck ratio of \(\psi\) by \(D_\psi(G) = \frac{\psi(G)}{\Sigma_{v\in V(G)}\psi(G-v)}\). We give generic upper and lower bounds on \(D_\psi\) for monotone increasing and monotone decreasing invariants \(\psi\), respectively.

Then, we proceed to consider the Wiener index \(W(G)\), showing that \(D_W(G) \leq \frac{1}{|V(G)|-2}\). We show that equality is attained for a graph \(G\) if and only if every induced \(P_3\) subgraph of \(G\) is contained in a \(C_4\) subgraph. Such graphs have been previously studied under the name of self-repairing graphs.

We show that a graph on \(n \geq 4\) vertices with at least \(\frac{n^2-3n+6}{2} – n + 3\) edges is necessarily a self-repairing graph and that this is the best possible result. We also show that a \(2\)-connected graph is self-repairing if and only if all factors in its Cartesian product decomposition are.

Finally, some open problems about the deck ratio and about self-repairing graphs are posed at the end of the paper.

For a graph \(X\) and a digraph \(D\), we define the \(\beta\) transformation of \(X\) and the \(\alpha\) transformation of \(D\) denoted by \(X^\beta\) and \(D^\alpha\), respectively.\(D^\alpha\) is defined as the bipartite graph with vertex set \(V(D) \times \{0,1\}\) and edge set \(\{((v_i,0), (v_j, 1)) \mid v_i v_j \in A(D)\}\).\(X^\beta\) is defined as the bipartite graph with vertex set \(V(X) \times \{0,1\}\) and edge set \(\{((v_i,0), (v_j, 1)) \mid v_i v_j \in A(X)\}\), where \(X\) is the associated digraph of \(X\).In this paper, we give the relation between the eigenvalues of the digraph \(D\) and the graph \(D^\alpha\) when the adjacency matrix of \(D\) is normal. Especially, we obtain the eigenvalues of \(D^\alpha\) when \(D\) is some special Cayley digraph.