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.

In this note it is shown that the number of cycles of a linear hypergraph is bounded below by its cyclomatic number.

The Padmakar-Ivan \((PI)\) index is a Wiener-Szeged-like topological index. In this paper, we study the \(PI\) index of thorn graphs, and we present a generally useful method which can reduce the computational amount of \(PI\) index strikingly.

The concept of the sum graph and integral sum graph were introduced by F. Harary. In this paper, we gain some upper and lower bounds on the sum number and the integral sum number of a graph and these bounds are sharp, and some new properties on the integral sum graph. Using these results, we could directly investigate and determine the exclusive integral sum numbers, the exclusive sum numbers, the sum numbers and the integral sum numbers of the graphs \(K_n\backslash E(2P_3)\), \(K_n\backslash E(P_3)\) and any graph \(H\) with minimum degree \(\delta(H) = n-2\) respectively as \(2\) is more than a given number. Then they will be the beginning of a new thought of research on the (exclusive) sum graph and the (exclusive) integral sum graph.

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 vertex set \(V(G)\) and edge set \(E(G)\), and let \(g\) and \(f\) be two integer-valued functions defined on \(V(G)\) such that \(0 \leq g(x) \leq f(x)\) for each \(x \in V(G)\). A \((g, f)\)-factor of \(G\) is a spanning subgraph \(F\) of \(G\) such that \(g(x) \leq d_F(x) \leq f(x)\) for each \(x \in V(F)\). A \((g, f)\)-factorization of \(G\) is a partition of \(E(G)\) into edge-disjoint \((g, f)\)-factors. Let \({F} = \{F_1, F_2, \ldots, F_m\}\) be a factorization of \(G\) and \(H\) be a subgraph of \(G\) with \(m\) edges. If \(F_i\), \(1 \leq i \leq m\), has exactly one edge in common with \(H\), we say that \({F}\) is orthogonal to \(H\). In this paper, it is proved that every \((mg+k-1, mf-k+1)\)-graph contains a subgraph \( {R}\) such that \( {R}\) has a \((g, f)\)-factorization orthogonal to any given subgraph with \(k\) edges of \(G\) if \(f(x) > g(x) \geq 0\) for each \(x \in V(G)\) and \(1 \leq k \leq m\), where \(m\) and \(k\) are two positive integers.

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 \(n \geq 3\).