We present some binomial identities for sums of the bivariate Fibonacci polynomials and for weighted sums of the usual Fibonacci polynomials with indices in arithmetic progression.

A complete arc of size \(q^2 – 1\) is constructed in the Moulton plane of order \(q^2\) for \(q \geq 5\) odd.

On the basis of the joint tree model initiated and comprehensively described by Liu, we obtain the genus distributions of double pear ladder graphs (a type of new \(3\)-regular graphs) in orientable surfaces.

The silicates are the largest, the most interesting and the most complicated class of minerals by far. The basic chemical unit of silicates is the \((\text{SiO}_4)\) tetrahedron. A silicate sheet is a ring of tetrahedrons which are linked by shared oxygen nodes to other rings in a two-dimensional plane that produces a sheet-like structure. We consider the silicate sheet as a fixed interconnection parallel architecture and call it a silicate network. We solve the Minimum Metric Dimension problem, which is NP-complete for general graphs.

A pebbling step on a graph consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. We consider all weight functions defined on the vertices of a graph that satisfy some property \({P}\). The \({P}\)-pebbling number of a graph is the minimum number of pebbles needed in an arbitrary initial configuration so that, for any such weight function, there is a sequence of pebbling moves at the end of which each vertex has at least as many pebbles as required by the weight function. Some natural properties on graph products are induced by properties defined on the factor graphs. In this paper, we give a bound for the \({P}’\)-pebbling number associated with a particular kind of product property \({P}’\) in terms of the \({P}_i\)-pebbling numbers associated with the factor properties \({P}_1\) and \({P}_2\). We do this by introducing color pebbling, which may be of interest in its own right.

The \(k\)-th isoperimetric edge connectivity \(\gamma_k(G) = \min\{|[U,\overline{U}]| : U \subset V(G), |U| \geq k\}\). A graph \(G\) with \(\gamma_k(G) = \beta_k(G)\) is said to be \(\gamma_k\)-optimal, where \(\beta_k(G) = \min\{|[U,\overline{U}]| : U \subset V(G), |U| = k\}\). Let \(G\) be a connected \(d\)-regular graph. Write \(L(G)\) and \(P_2(G)\) the line graph and the 2-path graph of \(G\), respectively. In this paper, we derive some sufficient conditions for \(L(G)\) and \(P_2(G)\) to be \(\gamma_k\)-optimal.

In 1968, Vizing conjectured that for any edge chromatic critical graph \(G = (V,E)\) with maximum degree \(\Delta\) and independence number \(\alpha(G)\), \(\alpha(G) \leq \frac{|V|}{2}\). This conjecture is still open. In this paper, we prove that \(\alpha(G) \leq \frac{3\Delta-2}{5\Delta-2}|V|\) for \(\Delta = 11, 12\) and \(\alpha(G) \leq \frac{11\Delta-30}{17\Delta-30}|V|\) for \(13 \leq \Delta \leq 29\). This improves the known bounds for \(\Delta \in \{11, 12, \ldots, 29\}\).

Consider a communication network \(G\) in which a limited number of edge (arc) and/or vertex faults \(F\) might occur. A routing \(\rho\), i.e. a fixed path between each pair of vertices, for the network must be chosen without knowing which components might become faulty. The diameter of the surviving route graph \(R(G, \rho)/F\), where \(R(G, \rho)/F\) is a digraph with the same vertices as \(G – F\) and a vertex \(x\) being adjacent to another vertex \(y\) if and only if \(\rho(x, y)\) avoids \(F\), could be an important measurement for the routing \(\rho\). In this paper, the authors consider the Cartesian product digraphs whose factors satisfy some given conditions and show that the diameter of the surviving route graph is bounded by three for any minimal routing \(\rho\) when the number of faults is less than some integer. This result is also useful for the Cartesian product graphs and generalizes some known results.

The Tribonacci Zeta functions are defined by \(\zeta_T(s) = \sum_{k=1}^{\infty} {T_{k}^{-s}}\). We discuss the partial infinite sum \(\sum_{n=1}^{\infty} {T_{k}^{-s}}\) for some positive integer \(n\). We also consider the continued fraction expansion including Tribonacci numbers.

Crossing numbers of graphs are in general very difficult to compute. There are several known exact results on the crossing numbers of Cartesian products of paths, cycles or stars with small graphs. In this paper we study \(\text{cr}(W_{1,m} \Box P_{n})\), the crossing number of Cartesian product \(W_{l,m} \Box P_{n}\), where \(W_{l,m}\) is the cone graph \(C_{m} + \overline{K_{l}}\). Klešč showed that \(\text{cr}(W_{1,3} \Box P_{n}) = 2n\) (Journal of Graph Theory, \(6(1994), 605-614)\)), \(\text{cr}(W_{1,4} \Box P_{n}) = 3n – 1\) and \(\text{cr}(W_{2,3} \Box P_{n}) = 4n\) (Discrete Mathematics, \(233(2001),353-359\)). Huang \(et\) \(al\). showed that \(\text{cr}(W_{1,m} \Box P_{n}) = (n – 1)\lfloor\frac{m}{2}\rfloor \lfloor\frac{m-1}{2}\rfloor +n+1\). for \(n \leq 3\) (Journal of Natural Science of Hunan Normal University,\(28(2005), 14-16)\). We extend these results and prove \(\text{cr}(W_{1,m} \Box P_{n}) = (n – 1) \left\lfloor \frac{m}{2} \right\rfloor\lfloor \frac{m-1}{2}\rfloor + n+1\) and \(\text{cr}(W_{2,m} \Box P_{n}) = 2n \left\lfloor \frac{m}{2} \right\rfloor\lfloor\frac{m-1}{2} \rfloor + 2n\).

A double-loop network (DLN) \(G(N;1,s)\) with \(1 < s < N\), is a digraph with the vertex set \(V = \{0,1,\ldots,N – 1\}\) and the edge set \(E=\{u\to v\mid v-u\equiv 1,s \pmod{N}, u,v \in V\}\). Let \(D(N;1,s)\) be the diameter of \(G\) and let us define \(D(N) = \min\{D(N;1,s)\mid 1 < s < N\}\) and \(lb(N) = \lceil\sqrt{3N}\rceil – 2\). A given DLN \(G(N;1,s)\) is called \(k\)-tight if \(D(N;1,s) = lb(N) + k\) (\(k \geq 0\)). A \(k\)-tight DLN is called optimal if \(D(N) = lb(N) + k\) (\(k \geq 0\)). It is known that finding \(k\)-tight optimal DLN is a difficult task as the value \(k\) increases. In this work, a practical algorithm is derived for finding \(k\)-tight optimal double-loop networks (\(k \geq 0\)), and it is proved that the average complexity to judge whether there exists a \(k\)-tight \(L\)-shaped tile with \(N\) nodes is \(O(k^2)\). As application examples, we give some \(9\)-tight optimal DLN and their infinite families.

Let \(k\) be a positive integer and let \(G = (V(G), E(G))\) be a graph with \(|V(G)| \geq 4k\). In this paper, it is proved that if the minimum degree sum is at least \(6k – 1\) for each pair of nonadjacent vertices in \(V(G)\), then \(G\) contains \(k\) vertex-disjoint chorded cycles. This result generalizes the main Theorem of Finkel. Moreover, the degree condition is sharp in general.

Let \(G = (V, E)\) be a finite simple connected graph. For any vertex \(v\) in \(V\), let \(N_G(v) = \{u \in V: uv \in E\}\) be the open neighbourhood of \(v\), and let \(N_G[v] = N_G(v) \cup \{v\}\) be the closed neighbourhood of \(v\). A connected graph \(G\) is said to be neighbourhood highly irregular (or simply NHI) if for any vertex \(v \in V\), any two distinct vertices in the open neighbourhood of \(v\) have distinct closed neighbourhood sets. In this paper, we give a necessary and sufficient condition for a graph to be NHI. For any \(n \geq 1\), we obtain a lower bound for the order of regular NHI graphs and a sharp lower bound for the order of NHI graphs with clique number \(n\), which is better than the bound attained earlier.

In this paper, we initiate the study of \(k\)-connected restrained domination in graphs. Let \(G = (V,E)\) be a graph. A \(k\)-connected restrained dominating set is a set \(S \subseteq V\) where \(S\) is a restrained dominating set and \(G[S]\) has at most \(k\) components. The \(k\)-connected restrained domination number of \(G\), denoted by \(\gamma_r^k(G)\), is the smallest cardinality of a \(k\)-connected restrained dominating set of \(G\). First, some exact values and sharp bounds for \(\gamma_r^k(G)\) are given in Section 2. Then, the necessary and sufficient conditions for \(\gamma_r(G) = \gamma_r^1(G) = \gamma_r^2(G)\) are given if \(G\) is a tree or a unicyclic graph in Section 3 and Section 4.

The first two authors have shown, in \([13]\), that if \(K_{r,r} \times K_{m}\), \(m \geq 3\), is an even regular graph, then it is Hamilton cycle decomposable, where \(\times\) denotes the tensor product of graphs. In this paper, it is shown that if \((K_{r,r} \times K_{m})^*\) is odd regular, then \((K_{r,r} \times K_{m})^*\) is directed Hamilton cycle decomposable, where \((K_{r,r} \times K_{m})^*\) denotes the symmetric digraph of \(K_{r,r} \times K_{m}\).

In \([8]\) the concept of \(H\)-kernel was introduced, which generalizes the concepts of kernel and kernel by monochromatic paths. In this paper, we prove necessary and sufficient conditions for the existence of H-kernels in the \(D\)-join of digraphs, and consequently, we will give a sufficient condition for the \(D\)-join to be \(H\)-kernel perfect.

Let \(\operatorname{MPT}(v,\lambda)\) denote a maximum packing of triples of order \(v\) with index \(\lambda\). For \(\lambda > 1\) and \(v \geq 3\), it is proved in this paper that the necessary and sufficient condition for the embedding of an \(\operatorname{MPT}(v,\lambda)\) in an \(\operatorname{MPT}(u,\lambda)\) is \(u \geq 20v + 1\).

The maximal and next-to-maximal subspaces of a nonsingular parabolic quadric \(Q(2n,2)\), \(n \geq 2\), which are not contained in a given hyperbolic quadric \(Q_+(2n-1,q) \subset Q(2n,q)\) define a sub near polygon \(\mathbb{I}_n\) of the dual polar space \(DQ(2n,2)\). It is known that every valuation of \(DQ(2n,2)\) induces a valuation of \(\mathbb{I}_n\). In this paper, we show that also the converse is true: every valuation of \(\mathbb{I}_n\) is induced by a valuation of \(DQ(2n,2)\). We will also study the structure of the valuations of \(\mathbb{I}_n\).

The (Laplacian) spectral radius of a graph is the maximum eigenvalue of its adjacency matrix (Laplacian matrix, respectively). Let \(\mathcal{G}(n,k)\) be the set of bipartite graphs with \(n\) vertices and \(k\) blocks. This paper gives a complete characterization for the extremal graph with the maximum spectral radius (Laplacian spectral radius, respectively) in \(\mathcal{G}(n, k)\).

In the paper “A note on the eigenvalues of graphs, Ars Combinatoria \(94 (2010), 221-227\)” by Lihua Feng and Guihai Yu, page 226, we have the following note.

In this paper, we show that among all connected graphs of order \(n\) with diameter \(D\), the graph \(G^*\) has maximal spectral radius, where \(G^*\) is obtained from \(K_{n-D} \bigvee \overline{K_2}\) by attaching two paths of order \(l_1\) and \(l_2\) to the two vertices \(u,v\) in \(\overline{K_2}\), respectively, and \(l_1 + l_2 = D-2\), \(|l_1 – l_2| \leq 1\).

P. Erdés and T. Gallai gave necessary and sufficient conditions for a sequence of non-negative integers to be graphic. Here,their result is generalized to multigraphs with a specified multiplicity. This both generalizes and provides a new proof of a result in the literature by Chungphaisan \([2].\)

Let \(u\) and \(v\) be two vertices in a graph \(G\). We say vertex \(u\) dominates vertex \(v\) if \(N(v) \subseteq N(u) \cup \{u\}\). If \(u\) dominates \(v\) or \(v\) dominates \(u\), then \(u\) and \(v\) are comparable. The Dilworth number of a graph \(G\), denoted \(\operatorname{Dil}(G)\), is the largest number of pairwise incomparable vertices in the graph \(G\). A graph \(G\) is called claw-free if \(G\) has no induced subgraph isomorphic to \(K_{1,3}\). It is shown that if \(G\) is a \(k\) (\(k \geq 3\)) – connected claw-free graph with \(\operatorname{Dil}(G) \leq 2k-5\), then \(G\) is Hamilton-connected and a Hamilton path between every two vertices in \(G\) can be found in polynomial time.

In this paper, we analyze the familiar straight insertion sort algorithm and quantify the deviation of the output from the correct sorted order if the outcomes of one or more comparisons are in error. The disarray in the output sequence is quantified by six measures. For input sequences whose length is large compared to the number of errors, a comparison is made between the robustness to errors of bubble sort and the robustness to errors of straight insertion sort. In addition to analyzing the behaviour of straight insertion sort, we review some inequalities among the various measures of disarray, and prove some new ones.

In this article, we give new lower bounds for the size of edge chromatic critical graphs with maximum degrees of \(8\) and \(9\), respectively. Furthermore, it implies that if \(G\) is a graph embeddable in a surface \(S\) with characteristics \(c(S) = -1\) or \(-2\), then \(G\) is class one if maximum degree \(\Delta \geq 8\) or \(9\), respectively.

While powers of the adjacency matrix of a finite graph reveal information about walks on the graph, they fail to distinguish closed walks from cycles. Using elements of an appropriate commutative, nilpotent-generated algebra, a “new” adjacency matrix \(\Lambda\) can be associated with a random graph on \(n\) vertices. Letting \(X_k\) denote the number of \(k\)-cycles occurring in a random graph, this algebra together with a probability mapping allow \(\mathbb{E}(X_k)\) to be recovered in terms of \(\operatorname{tr} \Lambda^k\). Higher moments of \(X_k\) can also be computed, and conditions are given for the existence of higher moments in growing sequences of random graphs by considering infinite-dimensional algebras. The algebras used can be embedded in algebras of fermion creation and annihilation operators, thereby establishing connections with quantum computing and quantum probability theory. In the framework of quantum probability, the nilpotent adjacency matrix of a finite graph is a quantum random variable whose \(m\)th moment corresponds to the \(m\)-cycles contained in the graph.

In \([2]\) it was introduced the concept of the kernel by monochromatic paths, which generalize concept of kernel. In this paper we prove the necessary and sufficient conditions for the existence of kernels by monochromatic paths in the \(D\)-join of digraphs. We also give sufficient condition for \(D\)-join to be monochromatic kernel perfect. The existence of generalized kernel (in distance sense) in D-join were studied in \([5]\). Moreover we calculate the total number of kernels by monochromatic paths in this product.

For integers \(p, q, s\) with \(p \geq q \geq 3\) and \(1 \leq s \leq q-1\), let \(\mathcal{K}^{-s}{p,q}\) (resp. \(\mathcal{K}_2^{-s}{p,q}\)) denote the set of connected (resp. 2-connected) bipartite graphs which can be obtained from \(K_{p,q}\) by deleting a set of \(s\) edges. In this paper, we prove that for any \(G \in \mathcal{K}_2^{-s}{p,q}\) with \(p \geq q \geq 3\), if \(9 \leq s \leq q-1\) and \(\Delta(G’) = s-3\) where \(G’ = K_{p,q} – G\), then \(G\) is chromatically unique.

Let \(k\) be a positive integer and \(G\) a graph with order \(n \geq 4k + 3\). It is proved that if the minimum degree sum of any two nonadjacent vertices is at least \(n + k\), then \(G\) contains a 2-factor with \(k + 1\) disjoint cycles \(C_1, \ldots, C_{k+1}\) such that \(C_i\) are chorded quadrilaterals for \(1 \leq i \leq k-1\) and the length of \(C_{k}\) is at most \(4\).

A finite simple graph is of class one if its edge chromatic number is equal to the maximum degree of this graph. It is proved here that every planar graph with the maximum degree \(5\) and without \(4\) or \(5\)-cycles is of class one. One of Zhou’s results is improved.

A cycle \(C\) in a graph \(G\) is said to be dominating if \(E(G-C) = 0\). Enomoto et al. showed that if \(G\) is a 2-connected triangle-free graph with \(\alpha(G) \leq 2\kappa(G) – 2\), then every longest cycle is dominating. But it is unknown whether the condition on the independence number is sharp. In this paper, we show that if \(G\) is a 2-connected triangle-free graph with \(\alpha(G) \leq 2\kappa(G) – 1\), then \(G\) has a longest cycle which is dominating. This condition is best possible.

In this paper, we obtain the explicit recurrences of the independence polynomials of polygonal cactus chains of two classes, and show that they are the extremal polygonal cactus chains with respect to the number of independent sets.

We prove that the power of cycles \(C_n^2\) for odd \(n\) are antimagic. We provide explicit constructions to demonstrate that all powers of cycles \(C_n^2\) for odd \(n\) are antimagic and their vertex sums form a set of successive integers.

A graph \(G = (V, E)\) is Skolem-graceful if its vertices can be labelled \(1, 2, \ldots, |V|\), so that the edges are labelled \(1, 2, \ldots, |E|\), where each edge label is the absolute difference of the labels of the two end-vertices. It is shown that a \(k\)-star is Skolem-graceful only if at least one star has even size or \(k \equiv 0\) or \(1 \pmod{4}\), and for \(k \leq 5\), a \(k\)-star is Skolem-graceful if at least one star has even size or \(k \equiv 0\) or \(1 \pmod{4}\). In this paper, we show that \(k\)-stars are Skolem-graceful if at least one star has even size or \(k \equiv 0\) or \(1 \pmod{4}\) for all positive integer \(k\).

Let \(\Gamma\) be a \(d\)-bounded distance-regular graph with diameter \(d \geq 3\) and with geometric parameters \((d, b, \alpha)\). Pick \(x \in V(\Gamma)\), and let \(P(x)\) be the set of all subspaces containing \(x\). Suppose \(P(x, m)\) is the set of all subspaces in \(P(x)\) with diameter \(m\), where \(1 \leq m < d\). Define a graph \(\Gamma'\) whose vertex-set is \(P(x, m)\), and in which \(\Delta_1\) is adjacent to \(\Delta_2\) if and only if \(d(\Delta_1 \cap \Delta_2) = m – 1\). We prove that \(\Gamma'\) is a distance-regular graph and compute its intersection numbers.

Let \(G\) be a \(contraction-critical\) \(\kappa\)-connected graph. It is known (see Graphs and Combinatorics, \(7 (1991) 15-21\)) that the minimum degree of \(G\) is at most \(\lfloor \frac{5\kappa}{4} \rfloor – 1\). In this paper, we show that if \(G\) has at most one vertex of degree \(\kappa\), then either \(G\) has a pair of adjacent vertices such that each of them has degree at most \(\lfloor \frac{5\kappa}{4} \rfloor – 1\), or there is a vertex of degree \(\kappa\) whose neighborhood has a vertex of degree at most \(\lfloor \frac{4\kappa}{4} \rfloor – 1\). Moreover, if the minimum degree of \(G\) equals to \(\frac{5\kappa}{4} – 1\) (and thus \(\kappa = 0 \mod 4\)), Su showed that \(G\) has \(\kappa\) vertices of degree \(\frac{5\kappa}{4} – 1\), guessed that \(G\) has \(\frac{3\kappa}{2}\) such vertices (see Combinatorics Graph Theory Algorithms and Application (Yousef Alavi et. al Eds.),World Scientific, \(1993, 329-337\)). Here, we verify that this is true.

A simple Kirkman packing design \(SKPD(\{w, w+1\}, v)\) with index \(\lambda\) is a resolvable packing with distinct blocks and maximum possible number of parallel classes, each containing \(u =v-w \lfloor \frac{v}{w} \rfloor\) blocks of size \(w+1\) and \(\frac{v-u(w+1)}{w}\) blocks of size \(w\), such that each pair of distinct elements occurs in at most \(\lambda\) blocks. In this paper, we solve the spectrum of simple Kirkman packing designs \(SKPD(\{3, 4\}, v)\) with index \(2\) completely.

In this paper, we study the matrices related to the idempotent number and the number of planted forests with \(k\) components on the vertex set \([n]\). As a result, the factorizations of these two matrices are obtained. Furthermore, the discussion goes to the generalized case. Some identities and recurrences involving these two special sequences are also derived from the corresponding matrix representations.

A Roman dominating function on a graph \(G = (V, E)\) is a function \(f : V \rightarrow \{0, 1, 2\}\) satisfying the condition that every vertex \(u\) for which \(f(u) = 0\) is adjacent to at least one vertex \(v\) for which \(f(v) = 2\). The weight of a Roman dominating function is the value \(f(V) = \sum_{u \in V} f(u)\). The minimum weight of a Roman dominating function on a graph \(G\), denoted by \(\gamma_R(G)\), is called the Roman domination number of \(G\). In [E.J. Cockayne, P.A. Dreyer, Jr.,S.M. Hedetniemi, S.T. Hedetniemi, Roman domination in graphs,Discrete Math. \(278(2004) 11-22.]\), the authors stated a proposition which characterized trees which satisfy \(\gamma_R(T) = \gamma(T) + 2\), where \(\gamma(T)\) is the domination number of \(T\). The authors thought the proof of the proposition was rather technical and chose to omit its proof; however, the proposition is actually incorrect. In this paper, we will give a counterexample of this proposition and introduce the correct characterization of a tree \(T\) with \(\gamma_R(T) = \gamma(T) + 2\).

Let \(G\) be a graph on \(2n\) vertices with minimum degree \(r\). We show that there exists a two-coloring of the vertices of \(G\) with colors \(-1\) and \(+1\), such that all open neighborhoods contain more \(+1\)’s than \(-1\)’s, and altogether the number of \(+1\)’s does not exceed the number of \(-1\)’s by more than \(O(\frac{n}{\sqrt{n}})\).

The \(Problème \;des \;Ménages\) \((Married \;Couples \;Problem)\), introduced by E. Lucas in 1891, is a classical problem that asks for the number of ways to arrange \(n\) couples around a circular table, such that husbands and wives are in alternate places and no couple is seated together. In this paper, we present a new version of the Menage Problem that carries constraints consistent with Muslim culture.

Let \(G\) be a connected simple graph with girth \(g\) and minimal degree \(\delta \geq 3\). If \(G\) is not up-embeddable, then, when \(G\) is 1-edge connected,

\[\gamma_M(G) \geq \frac{D_1(\delta,g)-2}{2D_1(\delta,g)-1}\beta(G)+ \frac{D_1(\delta,g)+1}{2D_1(\delta,g)-1}.\]

When \(G\) is \(k\)(\(k = 2, 3\))-edge connected ,

\[\gamma_M(G) \geq \frac{D_k(\delta,g)-1}{2D_k(\delta,g)}\beta(G)+ \frac{D_k(\delta,g)+1}{2D_k(\delta,g)}.\]

The functions \(D_k(\delta, g)\) (\(k = 1, 2, 3\)) are increasing functions on \(\delta\) and \(g\).

In this paper, the authors discuss the values of a class of generalized Euler numbers and generalized Bernoulli numbers at rational points.

For two vertices \(u\) and \(v\) in a graph \(G = (V,E)\), the detour distance \(D(u,v)\) is the length of a longest \(u-v\) path in \(G\). A \(u-v\) path of length \(D(u,v)\) is called a \(u-v\) detour. A set \(S \subseteq V\) is called a weak edge detour set if every edge in \(G\) has both its ends in \(S\) or it lies on a detour joining a pair of vertices of \(S\). The weak edge detour number \(dn_w(G)\) of \(G\) is the minimum order of its weak edge detour sets and any weak edge detour set of order \(dn_w(G)\) is a weak edge detour basis of \(G\). Certain general properties of these concepts are studied. The weak edge detour numbers of certain classes of graphs are determined. Its relationship with the detour diameter is discussed and it is proved that for each triple \(D, k, p\) of integers with \(8 \leq k \leq p-D+1\) and \(D \geq 3\) there is a connected graph \(G\) of order \(p\) with detour diameter \(D\) and \(dn_w(G) = k\). It is also proved that for any three positive integers \(a, b, k\) with \(k \geq 3\) and \(a \leq b \leq 2a\), there is a connected graph \(G\) with detour radius \(a\), detour diameter \(b\) and \(dn_w(G) = k\). Graphs \(G\) with detour diameter \(D \leq 4\) are characterized for \(dn_w(G) = p-1\) and \(dn_w^+(G) = p-2\) and trees with these numbers are characterized. A weak edge detour set \(S\), no proper subset of which is a weak edge detour set, is a minimal weak edge detour set. The upper weak edge detour number \(dn_w^+(G)\) of a graph \(G\) is the maximum cardinality of a minimal weak edge detour set of \(G\). It is shown that for every pair \(a, b\) of integers with \(2 \leq a \leq b\), there is a connected graph \(G\) with \(dn_w(G) = a\) and \(dn_w^+(G) = b\).

The vertex Padmakar-Ivan \((PI_v)\) index of a graph \(G\) is defined as the summation of the sums of \([m_{eu}(e|G) + m_{eu}(e|G)]\) over all edges \(e = uv\) of a connected graph \(G\), where \(m_{eu}(e|G)\) is the number of vertices of \(G\) lying closer to \(u\) than to \(v\), and \(m_{eu}(e|G)\) is the number of vertices of \(G\) lying closer to \(v\) than to \(u\). In this paper, we give the explicit expressions of the vertex PI indices of some sums of graphs.